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Actually, the bigger issue is Tefillin.

But first of all, while there is a standard practice to wear a Tallis Kattan (a smaller 4-cornered garment with Tzitzis, typically worn underneath a regular shirt) by all men, the Tallis Gadol (or just "Tallis") customs vary:

* Universally, men after they get married, even if they subsequently become divorced or widowed.
* Some groups start at Bar Mitzvah.
* While leading Davening (Shliach Tzibur). Generally for Shacharis (except possibly on Tisha B'Av), generally for all Davening on Shabbos/Yom Tov, but varies a bit by location for other times - e.g., never for weekday Mincha/Maariv, always for Mincha/Maariv, for Mincha only on Erev Shabbos/Yom Tov, only if you are not wearing a jacket, etc.
* When reading the Torah for a Minyan.
* When receiving an Aliyah in the Torah reading. Even this varies somewhat, with many places not requiring a Tallis for Mincha (Shabbos afternoon or fast days).

And now we get to Tisha B'Av. The customs on Tisha B'Av are closely related to those of mourning (Aveilus) in general. The most severe portion of mourning is the remainder of the day of the funeral. (From time of death until the funeral, there is no Davening, Brachos or positive Mitzvos done as the mourners are in a special status (Onen) where the only Mitzvah they are involved in is preparation for the funeral.) There is a general rule (there is some disagreement about the specifics under certain circumstances) that even though a mourner Davens following the funeral - typically Mincha but possibly even Shacharis if the funeral is early enough in the day, he does **NOT** put on Tefillin that day.

Tisha B'Av copies this rule to a limited degree. We don't get away with skipping Tefillin altogether - the level of mourning of Tisha B'Av never reaches quite that far. However, since the most serious part of the mourning is the morning, as included in other rules such as not sitting on regular chairs, men don't put on Tefillin in the morning and instead put them on for Mincha. Tallis simply follows along with the Tefillin rule, though as far as I know most men who wear a Tallis Kattan do so on the entire daytime of Tisha B'Av.

One of the reasons for wearing Tefillin specifically in the morning is that the first two paragraphs of the Shema (which is specifically said in the first 3 Halachic hours of the day) are included in the Tefillin. However, Tefillin can be worn at any time during the day and on Tisha B'Av we make an exception due to the special mourning of the day. Plus there are other days we say Shema without Tefillin (Shabbos, Yom Tov, Chol Hamoed for many) so they are related but not dependent upon each other.