Aiding the Inquisition
After Lavellan (the default surname of an elven Inquisitor) survives the explosion at the Conclave, she becomes the first suspect due to being the sole survivor, having a mysterious glowing green mark on her hand, and not remembering what happened. Cassandra Pentaghast and Leliana (the Right and Left Hands of the Divine, who was also killed at the Conclave) mourn Divine Justinia V and interrogate Lavellan over the latter's involvement. Cassandra takes Lavellan to the Temple of Sacred Ashes, the site of the explosion, to show her what happened and to give Lavellan a chance to prove her innocence.
On Lavellan's journey to the explosion site, she meets up with Cassandra's other contacts who are fighting off demons that have slipped through the tear in the Veil (the Breach). Solas is one of these contacts. He humbly introduces himself, choosing to not reveal his key role in making sure Lavellan survived a three-day state of unconsciousness. The other contact, Varric Tethras, is the one to bring up Solas's role.
Due to his studies on the Fade, Solas theorizes the mark on Lavellan's hand may have the power to control and close the rifts, since the mark reacts everytime the Breach expands. To test this theory, Cassandra gives Lavellan a chance to prove her innocence by going into the Temple of Sacred Ashes and closing the original rift there. At the Temple, Lavellan successfully battles demons and closes the rift, as well as learn some valuable clues on who attacked the Conclave and killed Justinia. After falling unconscious and waking up (again), Lavellan is regarded by the people of Haven—where the Temple of Sacred Ashes was near—as blessed and sent by the Maker. She is referred to as the Herald of Andraste, a title notably ironic if the player chose to be a Dalish elf, as the Dalish do not believe in the Maker or the Chantry.
Cassandra and Leliana follow Divine Justinia V's last order to establish an Inquisition to take care of the Mage-Templar War—which was a direct result of the events in Dragon Age II—but choose to focus the efforts in closing the rifts and fighting the forces behind it. As the only person with the power to close the rifts, Lavellan falls into a leadership role and travels across Thedas to both seal rifts and clean up the chaotic consequences of the Mage-Templar War.
As Lavellan continues to seal rifts, she inevitably crosses paths with Corypheus, the Tevinter Magister who caused the Breach. He launches a surprise attack on the Inquisition's base in Haven, resulting in Haven's decimation. Lavellan survives Corypheus's assault and, with the rest of the Inquisition, regroups to find a new home.
"You shamed him when you destroyed Haven. It spoiled his glorious victory. It would be worse to acknowledge that you had done so. He must continue on his course or show weakness."
Solas, for his part, volunteered to assist the Inquisition immediately after the explosion at the Temple of Sacred Ashes. He is near Haven when he sees the explosion and the resulting rift, choosing to willingly walk into camp and surrendering his staff. As an apostate, outside of the control of the Chantry Circles, it was at a considerable risk to himself. While Cassandra is understandably suspicious initially, his goodwill and cooperation earned her trust, and he and his knowledge on the Fade are instrumental in the fight against Corypheus and demons from the other side.
It is also Solas's knowledge on elven history that enables him to suggest Skyhold, an abandoned elven fortress, as the new Inquisition stronghold. After everyone settles in Skyhold, Lavellan is officially declared the Inquisitor, the head of the Inquisition. As an elven mage, you can choose to express your bewilderment by pointing out how unsual it was to pick a mage or an elven Inquisitor.
The selection of an elven Inquisitor is particularly poignant because of the Chantry's long history of erasing and oppressing elves, such as their removal of all Chantry records on the elven slave Shartan's role in Andraste's rebellion and cropping his ears in historical paintings. Being led by a mage Inquisitor is also poignant because of the historical oppression of mages. (I played an elven mage; what can I say? I like to be contentious.)
On the first opportunity during the journey to Skyhold, Solas will request a private conversation with Lavellan, and reveal to her that the Orb Corypheus used to create the Breach is of elven origin. This is the same Orb Lavellan touched by accident at the Conclave, when she stumbled upon Corypheus using Divine Justinia V as sacrifice, and interrupted the ritual and received the Anchor—the mark used to control rifts. Corypheus had intended it for himself so he could tear down the Veil and walk physically into the Fade and make himself a god; instead, Lavellan has been using it to close the rifts in the Veil.
With human suspicion and treatment of elves so dismal, Solas expressed his concern that this could lead to humans blaming elves, and hoped Lavellan will do her best to save Thedas, so that the humans will remember their part. As with everything Solas does, he is most interested in assisting the common people, and he views joining the Inquisition as the best position for himself to achieve that—which is why he volunteered. He studied the Fade and used his knowledge on the Fade (more on this in Rift Mage) to help the Inquisition defeat Corypheus.
Post reveal, you can see Solas's true motivation. While he does indeed want to seal the rifts and defeat Corypheus, it's less to do with saving Thedas and more to do with wanting to clean up the mess he indirectly created. After which he can progress with his own plans.
Regardless of true motivations, Solas is a key player in establishing the Inquisition, both in researching the rifts and in locating Skyhold as a base. His knowledge on the Fade, spirits, and demons also prove instrumental. He is very invested in the fight against Corypheus. Whereas some companions are optional recruits, or if their approval get low enough they may leave, Solas will stay on even if approval with him is very low. You even have the option of punching him in the face when you disagree, and it changes nothing.