Here's a breakdown:
* "The board met" is the main clause, indicating the action.
* "to consider the proposal" is the infinitive phrase, explaining why the board met.
The infinitive phrase acts as an adverbial modifier because it tells us why the board met.
You could rephrase the sentence to emphasize the purpose:
* The board met for the purpose of considering the proposal.
* The board met in order to consider the proposal.
In both cases, the infinitive phrase "to consider the proposal" is still expressing the reason for the meeting.