Πρέπει να βγάλω τον λεκέ από το πουκάμισο πριν το βάλω στο πλυντήριο.

Πρέπει means it’s necessary / one must / have to. In everyday speech it often corresponds to have to (practical necessity), but it can also sound like must depending on context and tone.
Grammatically, πρέπει is impersonal (literally “it is necessary”), and it’s followed by να + verb.

In Greek, after verbs/expressions of necessity, desire, possibility, etc., you typically use the subjunctive, which is marked by να.
So Πρέπει να βγάλω… = I have to (that) I-remove… → natural Greek for “I have to remove…”

βγάλω is the aorist subjunctive form of βγάζω. Greek often chooses between:

• Imperfective (continuous/repeated): να βγάζω = to be removing / to remove regularly
• Perfective (single complete action): να βγάλω = to remove (successfully, once)

Here you want a completed result: get the stain out, so να βγάλω fits.

The dictionary form is βγάζω (also seen as βγάζω / βγάλω in some resources because both stems are important).
Key forms:

• Present: (εγώ) βγάζω = I take out / I remove
• Aorist (completed): (εγώ) έβγαλα = I took out / I removed
• Subjunctive aorist: να βγάλω = (that) I remove (once, fully)

τον λεκέ is accusative masculine singular:

• ο λεκές = the stain (nominative)
• τον λεκέ = the stain (accusative, direct object)

It’s accusative because it’s the direct object of βγάλω (what you remove).

από means from / out of. With βγάζω it naturally expresses removing something from a place/surface:
βγάζω τον λεκέ από το πουκάμισο = I remove the stain from the shirt.

πουκάμισο is neuter, so it takes neuter articles:

• nominative/accusative: το πουκάμισο
• genitive: του πουκαμίσου (less common in everyday speech, but correct)

Neuter nouns use το in both subject and object positions.

το is a neuter object pronoun meaning it, and it refers back to το πουκάμισο (the shirt).
Greek commonly replaces a repeated noun with a clitic pronoun:

• …από το πουκάμισο πριν το βάλω…
  = …from the shirt before I put it…

In Greek, weak object pronouns (like το, τον, την, τα) normally come before the verb:

• το βάζω = I put it
• να το βάλω = to put it

So πριν το βάλω is the standard word order.

στο is just a contraction of σε + το:

• σε το πλυντήριο → στο πλυντήριο = in/into the washing machine

This contraction is extremely common in modern Greek (also στη, στον, στα, etc.).