What tragic historic events occurred during the 3 weeks between 17 Tamuz and 9 Av that warrants a mourning period? Also, why is there a more "severe" mourning period (i.e. - greater restrictions such as not eating meat) during the 9 days beginning from Rosh Hodesh Av until 9 Av?
The Three Weeks are those days between 17 Tamuz, when the walls around Yerushalaim were breached by Titus's army, and 9 Av, when the Bais Hamikdash was actually destroyed. It would have been a period of heavy fighting, more than just a siege.
As to your second question: The Gemara (Tanis) doesn't mention a mourning period starting from 17 Tamuz. It's an Ashkenazi minhag, and I believe is not held among the Sefardim. The Mishna in Tanis states
Mishenichnas Av m'ma'atin b'simcha, which is the source of the 9 Days. In addition, there are additional restrictions (in the Gemara) for
Shavua shechal bo, the week in which 9 Av falls.
I understand that the fighting may have intensified in the later period. There is mention of the Roman army entering Jerusalem on 7 Av, which begins a period of increased mourning (no siyum, etc.).