Learn Hardcore Hebrew: אני רוצה לקרוא ספר אחר בלילה.

Yes, in context you can sometimes drop אני. But in the present tense, Hebrew forms like רוצה show gender and number, not person, so רוצה by itself could mean something like want / wants depending on context. Using אני makes it clearly I.

Does רוצה tell me whether the speaker is male or female?

Not in ordinary unpointed spelling. Both the masculine רוֹצֶה and the feminine רוֹצָה are written רוצה without vowel marks.

So this written sentence could be said by either a man or a woman. You usually know from context or from hearing the pronunciation.

Why is לקרוא used after רוצה?

Because Hebrew normally uses רוצה + infinitive for want to do something.

So:

רוצה = want
לקרוא = to read

Together, רוצה לקרוא means want to read.

What does the ל־ at the beginning of לקרוא mean?