Strange, there is no entry for stýskat on neither English nor Czech Wiktionary... Is it a commonly used verb at all? There seems to be another verb postrádat which also means to miss. Then there is also chybět which we learnt in previous lessons... Which one of those is better to use in real life and is there any difference in them at all?
That's very difficult to do because English has no equivalent of the verb "stýskat se".
Here's an attempt: "It will itself to me after you nostalgize." Not every helpful, is it?
Probably better to try something less literal. "stýskat se" means to "miss someone" but the subject is not the person doing the "missing", nor the missed person, but instead an impersonal "it" - the feeling itself. The person struck by this feeling is expressed in the dative case, here it's "mi" (to me). The missed person is introduced with the preposition "po" in the locative case - "po tobě" (after you).
So...word by word: "Bude..." - it will, "se" - the reflexive pronoun that is a part of the verb, "mi" - to me (it's my feeling), "po tobě" - after you (you're the target of the feeling), "stýskat" - the mysterious verb referring to the feeling of longing, missing, or nostalgia = a derived noun "stesk".