What "poder atraparlo" is not
A dangling infinitive.
Section 26.7 of the Nueva Gramática shows examples like:
de haberlo sabido ella;
antes de llegar el cartero
Which are very similar to OP's sentence if we make the subject explicit:
antes de poder atraparlo yo.
These are called infinitive subordinates, and they are grammatically correct (definitely not "Tarzan speak" as some have said).
What "poder atraparlo" is, but has nothing to do whatsoever with the change in subject
A verbal periphrasis.
Indeed, "poder + verb" is one of the 140 periphrasis in Spanish, meaning "being able to" + verb. However, this fact is unrelated to the subject change you are asking about.
What the actual answer for the subject change is
A phenomenon called sujeto tácito (tacit subject) or sujeto elidido (elided subject), which is a form of ellipsis.
Yes, both the example and the translation are correct and, for native and non-native speakers alike, this can be confusing.
Take the following sentences:
El halcón estaba esperando a que el conejo saliese de su madriguera, pero se cansó de esperar.
El halcón salió volando antes de poder atraparlo.
Me fui acercando al halcón poco a poco con la red preparada, pero me oyó.
El halcón salió volando antes de poder atraparlo.
Yes, the two sentences in bold are exactly equal; and yet, their subjects and meaning are different.
The first one means "before the hawk could catch the rabbit":
antes de poder (el halcón) atraparlo
- subject (elided) = "el halcón"
- direct object = lo = "el conejo"
The second one means "before I could catch the hawk":
antes de poder (yo) atraparlo
- subject (elided) = "yo"
- direct object = lo = "el halcón"
The only way to tell the difference is the context.
- In the first example, the subject for poder atraparlo is taken from the preceding sentence, which says the hawk was waiting to catch the rabbit. So the subject (not) being able to catch must be the hawk, and the object is the rabbit.
- In the second example, the subject for poder atraparlo is taken from the preceding sentence too, which now says that I was waiting to catch the hawk with my net. So the subject (not) being able to catch must be me, and the object is the hawk.
This phenomenon is called sujeto tácito (also known as sujeto elidido), where the infinitive subordinate sentence does not have an explicit subject, and it must be inferred from the context. It does not have to be the same subject from the main sentence.
This is indeed a form of elipsis, a broader term which refers to the omission of part of a sentence, and the gist of it is the same in Spanish and English (see link above).
Notice that, without context, the subject could either be me, or us, or Elizabeth and her friends, or the hawk itself. Context is needed.
Is just happens that eliding the subject, especially when this subject is a pronoun, is way more common in Spanish compared to other languages (like English where the pronoun is mandatory) -- in fact, using "yo" at all in a sentence can be a telling sign that you are not a native speaker.
So, when the subject is elided and context is lacking, we tend to assume that the subject is the person speaking: thus "yo" (the one speaking) being assumed to be the subject of poder atraparlo, context not being enough to know otherwise.
Relevant sections of the Nueva Gramática: 26.7 and following; 33.4 and following; 41.11g; 26.11s; and some others referenced inside the book itself.