It is customary immediately after reciting the three blessings that comprise Birkot Ha'Torah to read verses from the Torah – specifically, the verses of Birkat Kohanim (the priestly blessing).
The reason for this practice is the opinion of the Rambam, that just as blessings over other Misvot must be immediately followed by the performance of the Misva, the blessing over the Misva of Torah learning must likewise be immediately followed by Torah learning. The Beracha over Tefillin, for example, is recited just prior to placing the Tefillin, and no interruption is allowed between the recitation of the Beracha and the placing of the Tefillin. By the same token, we must learn some Torah immediately after reciting Birkot Ha'Torah.
Others, however, disagree, and distinguish between Torah learning and other Misvot. They explain that Torah is always on a person's mind, even when he is not actually learning, as he intends to resume his learning as soon as he can. Therefore, one does not need to actually learn Torah immediately after reciting Birkot Ha'Torah, since his mind is always thinking about Torah.
In light of this difference of opinion, the Shulhan Aruch writes that one should preferably read verses from the Torah immediately after reciting Birkot Ha'Torah. If, however, one did not immediately read verses, he has nevertheless fulfilled his obligation and does not need to repeat Birkot Ha'Torah. In order to satisfy all opinions, the person should have in mind during the recitation of Ahabat Olam before Shema that this recitation should fulfill his Birkot Ha'Torah obligation.
As mentioned, the widespread custom is to read specifically the verses of Birkat Kohanim to fulfill this requirement to learn Torah after Birkot Ha'Torah. Different theories have been suggested to explain why these verses were chosen. The Eretz Tzvi (Rav Aryeh Tzvi Frommer of Kozhiglov, Poland, 1884-1943) writes that these verses were chosen because people know them by heart. Most people recite Birkot Ha'Torah from memory, without reading from a Siddur (despite the fact that it is preferable to read from a text), and so they will naturally read verses from memory afterward. Now Halacha does not allow reading verses from memory unless one knows them very well, with fluency. The custom therefore evolved to read after Birkot Ha'Torah the verses of Birkat Kohanim, with which people are very familiar, and may thus be read from memory.
Another explanation is that the verses of Birkat Kohanim contain 60 letters, corresponding to the 60 Masechtot (tractates) of the Talmud.
Some suggest that we recite the verses of Birkat Kohanim to draw our attention to the connection between the Kohanim and Torah study. The tribe of Levi – from which the Kohanim descend – did not participate in the sin of the golden calf, and then heeded Moshe's call to punish those who worshipped the calf, in reward for which this tribe's members were chosen to serve Hashem in the Bet Ha'mikdash, to bless the nation, and to teach Torah. In order to underscore this special role of the Kohanim, we recite the verses of Birkat Kohanim immediately following the blessings over the Misva of Torah study.
The Bet Yosef writes that during the times of the Bet Ha'mikdash, these verses were included in the special texts read by the "Mishmar" rotation each day, and so in commemoration, we read Birkat Kohanim every day.
If one read other verses instead of Birkat Kohanim, he has fulfilled his obligation, but of course it is proper to follow the accepted custom and read specifically verses of Birkat Kohanim.
Summary: Immediately after completing the recitation of Birkot Kohanim, one should read the verses of Birkat Kohanim. If one made an interruption before reading these verses, he does not repeat Birkot Ha'Torah. In order to satisfy all opinions, the person in this case should have in mind during the recitation of Ahabat Olam before Shema that this recitation should fulfill his Birkot Ha'Torah obligation.