How does the craving lead to lust, hatred and delusion?

How about rephrasing to: Ignorance leads to acts of lust, hatred, and delusion.

Craving is caused by attachment. If there is no attachment, there will be no more craving.

I think that "lust, hatred and delusion" are used as a very general (non-specific, imprecise) summary. In some suttas it's just two, i.e. "lust and delusion".

And "craving" is more specific or precise term, used for example in the description of the four noble truths, and descriptions of the twelve nidanas.

The (precise) term "craving" is related to, but distinguished from, "attachment" -- see for example Why do the Noble Truths talk about 'craving', instead of about 'attachment'?

In English-language, I think that "lust" sometimes means specifically sexual desire or lechery (reference).

But it can also mean desire more generally (reference):

Lust is a psychological force producing intense wanting for an object, or circumstance fulfilling the emotion. Lust can take any form such as the lust for sexuality, money or power. It can take such mundane forms as the lust for food as distinct from the need for food.

Wikipedia also describes it as a synonym for (a translation of) taṇhā (reference) -- which contradicts what you wrote in the question:

Lust holds a critical position in the philosophical underpinnings of Buddhist reality. It is named in the second of the Four Noble Truths, which are that

1. Suffering (dukkha) is inherent in all life.
2. Suffering is caused by lust.
3. There is a natural way to eliminate all suffering from one's life.
4. The Noble Eightfold Path is that way.

Pali words (like dukkha and taṇhā) are translated into English. The problem with that is:

• There may be no single exact word, maybe a good translation is a range of several words, but translators have to pick just one word (and different translators choose different words) -- so the translation is only approximate
• The English are used in other contexts, e.g. in (translations of) Christian doctrine, and therefore have acquired some meaning[s] that aren't present/inherent in the Buddhist/Pali words -- so the translatin is coloured or tainted with overloaded meaning

For these reasons if you want to discuss orthodox meanings (e.g. the ways in which words are used in the suttas) you may find it helpful or necessary to identify and discuss the Pali words.

If I tried to answer using only English then my answer would be based on something like "what does the work 'lust' mean to me, how do I interpret it, what do I associate it with", which may be too subjective or personal-opinion-based (and imprecise).

There are three types of craving: craving for sensual pleasures, craving to become something (that makes someone have ambition) and the craving to not become something (that makes one suicidal or withdrawn). From Itivuttaka 58:

This was said by the Blessed One, said by the Arahant, so I have heard: "There are these three cravings. Which three? Craving for sensuality, craving for becoming, craving for non-becoming. These are the three cravings."

If you crave for food, and eat some tasty food, you may cling to it or become attached to it. You think and fantasize about it even when it's not there. That's clinging or attachment. Craving gives rise to clinging and clinging gives rise to greed or lust to acquire it.

If someone prevents you from getting the tasty food that you cling to, then you feel angry. That hatred or aversion arises because you were denied what you cling to.

If this anger makes you enraged that you go and harm this person who prevents you from getting what you want, this is delusion. Delusion clouds your better judgment.

The craving to become something is similar. Examples are like wanting to get a gold medal or to get a promotion or to become doctor or professor or get recognition etc.

If you crave to get recognition and get it once, you may cling to it and want it even when it's not present. This leads you to do things to get recognition as you have greed (or lust) for it.

If you don't get recognition, but someone else does, you become envious. That's aversion.

If continuously not getting recognition leads you to become depressed, which clouds your normal healthy state of mind, then that's delusion.

According to AN 3.68:

The Blessed One said, "Monks, if you are asked by wanderers of other sects, 'Friends, there are these three qualities. Which three? Passion, aversion, & delusion. These are the three qualities. Now what is the difference, what the distinction, what the distinguishing factor among these three qualities?' — when thus asked, you should answer those wanderers of other sects in this way, 'Friends, passion carries little blame and is slow to fade. Aversion carries great blame and is quick to fade. Delusion carries great blame and is slow to fade.

"[Then if they ask,] 'But what, friends, is the reason, what the cause, why unarisen passion arises, or arisen passion tends to growth & abundance?' 'The theme of the attractive,' it should be said. 'For one who attends inappropriately to the theme of the attractive, unarisen passion arises and arisen passion tends to growth & abundance...'

"[Then if they ask,] 'But what, friends, is the reason, what the cause, why unarisen aversion arises, or arisen aversion tends to growth & abundance?' 'The theme of irritation,' it should be said. 'For one who attends inappropriately to the theme of irritation, unarisen aversion arises and arisen aversion tends to growth & abundance...'

"[Then if they ask,] 'But what, friends, is the reason, what the cause, why unarisen delusion arises, or arisen delusion tends to growth & abundance?' 'Inappropriate attention,' it should be said. 'For one who attends inappropriately, unarisen delusion arises and arisen delusion tends to growth & abundance...'

So, basically, you have cravings due to sensual pleasures and cravings to become something. When you give into the attraction towards it, greed or lust arises. When you give into irritation, typically because you are denied it, aversion arises.

When you pay too much attention to something that you shouldn't, like becoming depressed due to lack of recognition or distraught due to loss of a family member, or anxiety due to being worried about what other people think (which started from an aversion), or becoming enraged with others who criticized you, or having vengence due to thinking about what someone did to you a long time ago - these are all examples of delusion.

Firstly, you lust after something, due to craving and clinging. Secondly, being denied something you lust after, or losing it prematurely, would cause aversion. Thirdly, when you pay too much attention to your aversions (and also greed), it causes delusion.

Whatever experience we get used to and grow comfortable in -- with that we identify, to that we attach. Then anything that looks like it could help us get that experience again -- seems good to us, and anything that goes contrary to that -- seems bad to us.

Next time we see a sign, we recognize the form, we associate the name with that past experience. Based on that, we like or dislike the object. Whenever an object reminds of a good past experience, we crave that object because we crave that experience and we assume it will help us get there.

If you look at your mind very carefully, you'll see that it's not the object that you crave, it's a vague promise to get an experience. Lust/hatred/delusion is about confusing (conflating) experience with objects. You project your own memories onto objects, and based on that you form attitudes to objects.

For example, imagine you sit and suddenly want those sunglasses you've seen in the store yesterday. You liked how they looked on you, or, rather, you liked how you looked in the mirror. You liked how you looked, because your reflection looked cool. It looked cool, because it resembled the images of men you've seen in the advertis*m*nt. You liked those men because they were surrounded by the smiling women. Smiling women are happy women. You too want to be surrounded by the happy women. You want to be near women that laugh and have fun, because you want to be part of that experience. You crave the positive emotions of being in a company of happy women. You crave positive emotions and that's why you now sit and want those sunglasses!

This is how craving (for positive emotions) leads to lust (for an object).

All of this comes from naive generalization of objects and naive generalization of experiences. Naive generalization comes from ignorance. Like a magic trick, it only works if you don't look carefully. When you look carefully, it disbands:

Upon careful examination, these sunglasses don't really make you look like an actor in that ad. You do look somewhat similar but only under a certain lighting conditions when seen at a certain angle. The women in the ad do smile but are not actually happy, their smiles are smiles of professional models. You don't actually crave to be among the happy women, instead you are craving one experience when you were a little boy visiting your grandma in the summer, and your older cousin-sister and her friends used to play with you. You miss that experience of careless summer days and being playfully cared for by the older girls. More precisely, you miss the experience of being completely at ease, in a safe environment, someone more powerful taking care of you -- and buying those $150 sunglasses won't get you an inch closer to that. Worse than that, the "perfect" childhood experience of being safe and cared for was a 100% illusion because the six-years-old girls could not possibly take care of the four-years-old boy should any real situation arise; he only felt safe and at ease because of his ignorance!

Just like that, you sit on your cushion meditating, and you suddenly realize that you're feeling (subtle) suffering because you don't have $150 to buy those sunglasses. And then you realize that the entire situation, from sunglasses to the childhood memory of feeling at ease, is an illusory bubble due to a chain of naive generalizations! Once you see it completely and thoroughly, the illusion disbands like a dream and with it disappears all craving and all suffering associated with that.

So ignorance leads to naive generalization (formations), which leads to craving for experience (feeling), which leads to pursuit of objects (clinging), which leads to suffering.

Cessation of ignorance (seeing through the illusory bubble to its individual components) leads to cessation of naive generalizations (formations), which leads to cessation of craving for experience (feeling), which leads to cessation of the pursuit of objects (clinging), which leads to cessation of suffering.