Learn Turkish: Müdürle birlikte toplantıya giriyoruz. - We are entering the meeting together with the manager.

You can use either! The word ile (with) is very often attached directly to the previous word as a suffix. When attached, the i drops, and the suffix changes to match the word's vowels (using A-type vowel harmony). Since the last vowel in müdür is ü, it takes the -le form.

If the -le in müdürle already means "with," why do we need the word birlikte (together)?

You don't strictly need it, but adding birlikte emphasizes that you are doing the action together with someone. The combination -(y)la/-(y)le birlikte is a very common natural phrase in Turkish.

Why is it toplantıya and not just toplantı?

The verb girmek (to enter) requires you to show the direction or destination you are going towards. In Turkish, you must use the dative case (the "to/towards" suffix -a or -e) for this. So you are literally saying "We are entering to the meeting."

Where does the -y- come from in toplantıya?

The root word is toplantı (meeting), which ends in a vowel. The "to" suffix is also a vowel (-a). Because Turkish usually prevents two vowels from sitting right next to each other, the buffer letter -y- is inserted to connect them smoothly: toplantı-y-a.