The explanation is certainly correct in infinitive being non-finite form of a verb used with or without (to),eg:-let him stay ; ask him to stay Now to put=infinitive which relates to (to) stay=stay where (to) is ommited.Now also "stay in a cage with other animals" is treated as a clause extention of the sentence".NB Quotation marks should be noted."
For your reference only:-(2)participial phrase Or phrase containing a gerund:-staying.
so it avoids having two verbs in a sentence;
(3)asked----->stayed are all in the sequence of tense =the past tense.
are another possible correct usage of tense in English.