The two instances you indicated in the image are not
[VERB]+り. These are instead the verb in the -masu stem form, also sometimes called tine "continuative", "infinitive", or even "gerund" form in English, and also called the 連用形【れんようけい】 in Japanese.
The two verbs in your image:
Sometimes spelled in kanji as 取る, meaning "to take". This has the -masu form とります, and we see that とり is the stem form.
There are various threads on the site regarding the use of the 連用形【れんようけい】 to end a clause, as we see with the 「とり、」 example. In short, this is often roughly equivalent to saying "
[VERB], and..." in English.
Usually means "to enter, to go into something". This has the -masu form 入【はい】ります, and we see that 入【はい】り is the stem form.
However, in this specific case, the verb is part of a set phrase. The verb form of this set phrase is 気【き】に入【い】る, which literally means something like "to enter into one's mood", but it's used idiomatically to mean "to really like something". The noun form of this set phrase as used in your sample text is お気【き】に入【い】り, meaning basically "something one really likes, one's favorite".
The honorific お prefix here is a clue that the narrator isn't talking about themselves. Despite the honorific お prefix, this term has been fully lexicalized (i.e. it's become a word unto itself), and it is often used in reference to oneself as well as others.
In context, we have 彼【かれ】のお気【き】に入【い】りの歌手【かしゅ】の名前【なまえ】 or "the name of his favorite singers".
Please comment if the above does not answer your question.