Vi låner to bøker og legger dem i våre kofferter.

låner is the present‐tense form of the verb låne (to borrow). Norwegian verbs in group 1 (infinitive ends in –e) form the present by dropping the –e and adding –er:
låne → lån + er = låner
Present‐tense conjugation: jeg låner, du låner, han/hun låner, vi låner, dere låner, de låner.

legger is the present tense of legge (to put). Verbs ending in a single vowel + consonant often double the consonant before adding –er:
legg + er = legger
Conjugation: jeg legger, du legger, han/hun legger, vi legger, dere legger, de legger.

When two verbs share the same subject in coordinated clauses (joined by og), you can omit the subject pronoun in the second clause. So
Vi låner to bøker og legger dem…
means “We borrow two books and put them…” with vi understood before legger.

de is the subject pronoun “they,” while dem is the object pronoun “them.” Since the borrowed books (bøker) are the direct object of legger, you use dem:
legger dem = “put them.”

bok is a strong feminine noun with an irregular plural. Many feminine nouns form the indefinite plural by changing the vowel and adding –er:
bok (sg) → bøker (pl)
Definite forms are boka (sg def) and bøkene (pl def).

Possessive pronouns agree with the number of the noun. kofferter is plural, so you use våre (our–plural), not the singular vårt. Also, våre kofferter is indefinite (“our suitcases” in general). If you meant “in our specific suitcases,” you could say:
i koffertene våre