Jesus ate fish and honey as proof that he is not a spirit but still has a body of flesh. The food was real food and that is it. Jesus showed them that he is in the flesh just like them. Important fact is that he showed them that his body of flesh will ascend to heaven. On the new earth the lion will lie with the lamb! Are they spirits? nope. real fleshly animals. Like we will be.

I can respond to this question from the perspective of the theologian Emanuel Swedenborg (1688–1772) and the "New Church" denominations that follow his teachings.

The "natural" or physical nature of Jesus

First, in the story told in Luke 24:36–43, much of the focus is on Jesus establishing that he is not a spirit (i.e., a ghost), but is really, physically present with his disciples after his resurrection. This is why he calls special attention to his hands and feet (the outermost parts of the body), and also requests food so that he can eat something in their presence.

Since the issue concerns the natural, physical nature of Jesus, and he is establishing with them that he is physically, and not only spiritually, present with them, we would expect that the broiled fish and honeycomb would also relate to outward, natural, and physical things.

And this is precisely where Swedenborg says their significance lies.

The significance of fish and honey

In brief, Swedenborg says:

• Fish signify natural-level information and truth that we learn through our senses, from external sources, that play a part in drawing us toward Christ and salvation. (We'll deal with the "broiled" part later.)
• Honey (and honeycomb) signify natural-level good, meaning especially simple deeds of kindness and service to one another, and the delight or enjoyment we gain from them (signified by the sweetness of the honey).

We can see the significance of the fish and honey represented in this very story:

• The disciples required the assurance of their senses that Jesus had really risen from the dead, and was not just a ghost. They needed to see that he had solid (not just ghostly) hands and feet, and that he could eat real, physical food. So the fish that they gave him signified their need for natural-level information and truth about the risen Jesus Christ; the assurance of their own senses that he really was alive and with them, and not just some ephemeral ghost.
• In asking for food, Jesus asked the disciples to engage in one of the most common ways we humans serve each other and do natural, physical deeds of kindness for one another: we provide food and nourishment for those who are hungry. So the honeycomb that they gave him signified their enjoyment of that simple deed of kindness in giving him food, which also provided them assurance that he was very real and very much present with them.

A more specific meaning shown by the verses that follow

When looking for the significance of various people, items, and events in particular verses of the Bible, to get the full picture it is necessary to pay attention to the context that surrounds that verse.

In this instance, the very next verses speak of the Scriptures, their fulfillment in Jesus, and the disciples' understanding of them:

Then He said to them, "These are the words which I spoke to you while I was still with you, that all things must be fulfilled which were written in the Law of Moses and the Prophets and the Psalms concerning Me." And He opened their understanding, that they might comprehend the Scriptures. (Luke 24:44–45)

This, then, provides a clue that the fish and honeycomb have an even more specific meaning in this context: They relate to natural, external truth and goodness that are conveyed to us by the Scriptures, or the Word of God. Or in plain language, they represent the external, literal meaning of the Bible, and the understanding and goodwill that we receive from it.

Here is how Swedenborg stated this:

The honeycomb and the broiled fish which after His resurrection the Lord ate in the presence of the disciples was also a sign of the external sense of the Word, "the fish" meaning the truth associated with that sense and "the honeycomb" the pleasure attached to it. (Arcana Coelestia #5620:14)

He then quotes Luke 24:41–43, in which Jesus requests food and eats the offered broiled fish and honeycomb in the disciples' presence. And Swedenborg points out that the Lord next goes on to speak of the Scriptures and their fulfillment, giving this more specific meaning to the fish and honeycomb.

He is aware that this meaning may seem farfetched. He goes no to say:

The appearance is that nothing of the sort is meant, for it seems to have been purely by chance that they had part of a broiled fish and a honeycomb. But in fact their possession of these was providential—as is not only this but every other smallest fact mentioned in the Word. Because matters such as have been described were indeed meant, the Lord therefore referred to the Word, declaring that the things written in it had reference to Himself. But the things which have been written in the Old Testament Word regarding the Lord are but few in the sense of the letter, whereas everything contained in the internal sense has to do with Him; and it is from this that the Word gets its holiness. Everything contained in the internal sense is what is meant in the statement that "all things must be fulfilled which were written in the law of Moses, and the Prophets, and the Psalms concerning Him." (Arcana Coelestia #5620:14)

In other words, it was no accident that the particular food they happened to have was broiled fish and honeycomb. This was provided for through the Divine Providence that oversaw everything that takes place in the Bible. And though the fish and honeycomb relate specifically to the literal sense of the Bible—meaning the plain meaning of the stories and teachings as we read them in the Bible—the presence of these two particular foods was also provided by God for the sake of the deeper, spiritual meaning that lies within that literal sense of the Bible.

So the more specific meaning is that Jesus, as the Word made flesh (John 1:14), was instructing his disciples about the prophecies relating to him in the literal sense of the Scriptures (represented by the fish) and the joy they experienced not only in realizing that he was risen and truly alive, but in knowing that he was indeed the one prophesied in their Scriptures (the joy and delight being represented by the honeycomb).

Why broiled fish?

But there is one other detail in the story that is easy to miss. The fish was broiled, meaning cooked over a fire. On this significant detail, Swedenborg says:

"Roasted with fire" means good that is the product of love. This is clear from the meaning of "what is roasted with fire" as the good of love; for "fire" means love, . . . and "what is roasted" that which has been infused with love, consequently good. In the Word what has been roasted is distinguished from what has been boiled. "What has been roasted" means good, because it has been cooked by means of fire, while "what has been boiled" is used to mean truth, because it has been cooked by means of water. . . .

All this shows what "the roasted fish" in Luke 24:42—43 means in the spiritual sense, and also "the fish placed over the fire of coals" when the Lord appeared to the disciples, described in John as follows:

After the disciples got down onto the land they saw a fire of coals that had been set, and a small fish lying over it, and bread. Jesus came and took the bread and gave it to them, and did the same with the small fish. (John 21:9, 13)

"A fish" means truth in the natural, . . . while "a fire of coals" means good. Thus "a small fish lying over it" means the truth of spiritual good within the natural. (Arcana Coelestia #7852)

In contemporary language, the fact that the fish was broiled or roasted with fire rather than boiled or cooked in water signifies that when Jesus taught the disciples about the Scriptures that testified of him, he did it, not from some cold, intellectual stance, but rather out of the warmth of his love for them (signified by the fire that roasted the fish). Because he loved them, he wanted them to understand everything that the Scriptures said about them, both for their own eternal good and for the salvation they would bring to many others in the future who would hear their preaching and teaching, and experience their healing.

Summary

In this passage, then, in contemporary language, the two items Jesus ate signify:

• Broiled fish: Jesus, out of his love for all people, teaching the disciples—and us—the basic, simple truth of his real and substantial presence with us through opening their—and our—eyes to his presence in the narratives and teachings of the Word of God.
• Honeycomb: The joy, delight, and goodness we experience in seeing the Lord Jesus Christ present in his Word, and also present with us personally to lead us and guide us to eternal life through the love and goodness of his presence.

For further reference:

For those who want to read more of the source material behind this interpretation of the significance of the broiled fish and honeycomb, here are a few further references, in addition to the ones already provided in the answer above.

Be aware that these are from rather archaic translations (the only translations currently available for these works). Also, for those unfamiliar with Swedenborg's language, it can sound rather technical. Keep in mind that these works were written over two centuries ago.

To read the relevant material, at the linked page scroll down to the subsection number given after the main section number in the references below, which will be formatted there in brackets—for example: [19].

• Apocalypse Explained #513:19
• Apocalypse Explained #619:15
• Doctrine of the Lord #35:9–10

St Cyril points out that Song of Songs 5:1 is a for telling of the death and resurrection: "I came into the garden," referencing the grave, and, "I harvested my myrrh with my spices." These things point to the death, then he says, "I ate my bread with my honey." This is a prophesy of Jesus eating after his resurrection. (The word "bread" in that time was used for all manners of food.)

Dont read to much into it. Simple is always right, then build from there. Look at the accounts of the last supper in Luke 22:15-29, Mark 14:24-25, and Matthew 26:27-29. Jesus said you would not see him eating and drinking until he would again in the Kingdom of God. He did it to prove that the Kingdom had arrived. We are not waiting on the kingdom, but it is here already. Jesus came to restore God's kingdom to earth. Reference also Acts 10:41, Matthew 16:28, and Mark 9:1, 9:27. We are heaven's citizens now, ambassadors to influence earth now. Luke 24:35-48.

Some commentators do employ a bit of speculation on this question, including Augustine himself. The Ancient Christian Commentary on Scripture summarizes Augustine, saying that the fish "represent the faith of the martyrs that have gone through the fiery trials of suffering." With respect to a similar passage in John 21, Augustine writes that "[t]he fish roasted is Christ having suffered" (Treatise 123).

With respect to the honeycomb (which some manuscripts do not mention), Jamieson, Fausset, and Brown write that it is simply "common frugal fare" (Commentary). Matthew Henry likewise calls it "mean" (in the sense of "humble") and suggests that a lesson can be drawn from its inclusion:

The honey-comb, perhaps, was used as sauce to the broiled fish, for Canaan was a land flowing with honey. This was mean fare; yet, if it be the fare of the disciples, their Master will fare as they do, because in the kingdom of our Father they shall fare as he does, shall eat and drink with him in his kingdom. (Commentary)

The more important point, attested by both church fathers and more modern commentators, is that Jesus is proving to his disciples that he is truly resurrected and that he has a real body. As John Calvin writes:

During the whole course of his life, he had subjected himself to the necessity of eating and drinking; and now, though relieved from that necessity, he eats for the purpose of convincing his disciples of the certainty of his resurrection.