‘Au moment de ….’ - after prepositions in a preceding non-verbal expression, the verb form will be in the infinitive, and doesn’t strictly ‘have a tense’ in its own right

https://www.lawlessfrench.com/grammar/infinitive/

https://www.thoughtco.com/french-infinitive-linfinitif-1368866 

For the second part, the “ l’ “ is the direct object pronoun “la” elided to l’ in front of the vowel,  playing the same role in French as the pronoun ‘it’ does in English, which in this case is representing ‘la veste’ or ‘the jacket’. In French ‘ … enfiler la veste …‘ becomes ‘ … l’enfiler … ‘ with the object pronoun used instead of the noun, and preceding the verb as usual. 

(Of course, with a masculine noun ‘ l’ ‘ would still be used, but represent ‘le…’)