The film's official novelisation goes into more detail about Anakin's desires. In short, the loss of his mother and his illicit relationship with with Padmé has driven him close to madness. His prophetic dreams of her death have left him with a borderline pathological ambition to save her, regardless of the cost. He's of the opinion that having access to the secret holocrons of the Jedi will give him the ability to do this.
Palpatine had somehow seen into his secret heart, and had chosen to offer him the one thing he most desired in all the galaxy. He didn’t care about the Council, not really-that was a childish dream. He didn’t need the Council. He didn’t need recognition, and he didn’t need respect. What he needed was the rank itself.
All that mattered was Mastery.
All that mattered was Padme.
This was a gift beyond gifts: as a Master, he could access those forbidden holocrons in the restricted vault.
He could find a way to save her from his dream …
He shook himself back to the present. “I … am overwhelmed, sir. But the Council elects its own members. They will never accept this.”
For the record, he's not lying to Obi-Wan. He didn't ask for a seat on the council, here merely voiced his frustration to Palpatine that he wasn't on the council despite being so puissant.