How do sedevacantists who claim the See has been empty since Vatican II understand the legitimacy of the bishops and priests?

I intend to be concise, not abrasive; therefore, please do not take my tone as anything other than direct.

Distinctions and terms matter.

I was recently listening to my favorite Catholic...

According to whom/what? That is implicit though indirect question-begging; your bias is showing. If "The Sedevacantists" are correct, then Mr Horn, as a so-called Apologist, the entity he represents and defends, and his methods are not Catholic.

In other words, you tacitly assume that he and his "Church" are correct from the outset. Why bother asking your question then? (See "Reflex principle" for a germ of a possible, sound response.)

... becoming a priest/bishop required the laying of hands by other legitimate... priests/bishops...

False premise: it does not. Please make the distinction between sacramental validity and liceity, i.e., "legitimacy." In other words, conferring (valid) holy orders, licit or otherwise requires at least one validly consecrated bishop, not a licit or even Catholic one.

We may easily demonstrate this from, for example, two facts:

1. Even Trent Horn acknowledges that the so-called Eastern Orthodox have always had valid sacraments. However, at least up to "Vatican 2," the Church did (and does) not consider them licit, as She (The Church) thinks the "Eastern Orthodox" schismatic. (See 1054 AD)
2. The (Catholic) Church, again at least pre-"Vatican 2", formally declared that the Anglican orders are invalid. (See, e.g., https://www.newadvent.org/cathen/01491a.htm)

As at least one premise is false, the rest does not follow. Thus, this question is DoA.

How would a Sedevacantists respond to this?

Sadly, we are not monolithic, which would follow in an instance of, for example, a vacancy of the Holy See.

I hope that helps. At least I am a Sedevacantist. First-hand beats second, yes?

I advise you to go to vaticancatholic.com with further questions.

Sedevacantists say that there is a "bishop in the woods" still out there that was consecrated by what they think is the last valid pope (usually Pius XII). Supposedly this saves the visibility of the Church and apostolic succession, but they can't identify who this last valid bishop is.

Currently, the oldest bishops in the world seem consecrated no earlier than 1961, under John XIII (e.g., Bishop José de Jesús Sahagún de la Parra).

One of the last Pius XII-consecrated bishops, Dominik Kalata, S.J. (consecrated 1955), passed away 23 August 2018. Manuel José Bernardino Piñera Carvallo (consecrated 27 April 1958) died 21 June 2020. Pius XII died 9 October 1958.

See "'Bishop in the Woods' Theory", pp. 69-72 of True or False Pope? Refuting Sedevacantism and Other Modern Errors. Cf. also: Sedevacantism: A False Solution to a Real Problem.

Related:

• Crisis Series #34 w/ Don Tranquillo: Is Sedevacantism the Answer to the Crisis?
• Papal Heresy and Loss of Office - SSPX Interview Series

The following describes the position of most sedevacantists (as far as I know), including me. There may well be other sedevacantists who hold the theory described by Geremia, and there are also some who hold some theories about being able to elect a pope on their own.

Bishops consecrated before the current situation (sede vacante) began have consecrated more bishops since then, some of whom have consecrated yet more bishops. (For example, Archbishop Marcel Lefebvre consecrated four bishops in 1988. Archbishop Ngo Dinh Thuc also consecrated new bishops.) All these bishops can not only consecrate more bishops when needed but can also ordain priests. So there is no problem about having validly consecrated bishops and validly ordained priests.

As far as I know, the new church considers these consecrations valid but illicit, because they lack papal permission. Sedevacantists and some other traditionalists consider them licit because of the need to provide the sacraments for the faithful and the impossibility of obtaining papal permission. (Even those traditionalists who consider Bergoglio a valid pope realize that there's no chance of his permitting consecration of a bishop who fully adheres to the Catholic faith, i.e., pre-Vatican II.)