First of all, the possible answers to this question will be based on context clues and making inferences. The actual motive of the bomber is never revealed in the story. Secondly, the question could have a different answer depending on what point in the story it is asked.
The first bomb goes off in the Theodorakis coffee shop, making a large mess but essentially injuring no one. The second bomb goes off in the kitchen of Hoo's restaurant. This time Sydelle Pulaksi (Angela's partner) is minorly injured. At this point in the game, the identity of the bomber is unknown, but the characters themselves suggest various motives. Some believe it is Westing's murderer, attempting to strike again (like the will suggests). Others believe it is simply someone playing innocent pranks, trying to further rile up the already nervous players. Along this line, if the bomber is in fact an heir, another motive could be to throw the others off on their journey toward a solution.
The third bomb, however, is more revealing. It is hidden among the presents at Angela Wexler's wedding shower, and goes off in her face, sending her to the hospital with a facial cut that will leave a scar and burned hands. At this point, the bomber's motive most certainly seems to be more than an innocent prank.
The reader finds out at the end of chapter 16 that the bomber is actually Angela Wexler herself. This ironic twist proposes several possible motives. Perhaps Angela, who is constantly praised for her beauty (not brains) and marrying a man she is not ready to marry, is looking for a way to change the course of her life and do something she truly wishes to do, which is go to college. Her time of healing in the hospital affords her the opportunity to think through future decisions and she seems perfectly content to have a scar on her beautiful face for the rest of her life. Perhaps she is looking to shake up a life that she is currently finding mundane and inescapable.
Another motive might be that she is looking for attention in a different way, but does not know how to get it. Only her sister is aware that she is the bomber, so as a motive, this one is completely personal. It is unlikely that her motive was to actually (seriously) hurt someone, including herself. It is possible, however, that she was trying to scare the others.
The final bomb goes off in the elevator with Turtle inside, singing one of her braids and causing pain "too great to be put into words" (129). This one is set by Turtle, and she confesses to it. Her motive is to take the blame (for all the bombs) so that no one suspects her sister Angela.
Whatever her motive for the bombs, we know at the end of the story that Angela Wexler cancels her wedding and goes through college and med school to become a surgeon. It is clear that she was very confused, unhappy, and afraid to speak up for herself during the time of the Westing game. In the end, she ends up marrying Denton Deere several years down the road, so her love for him was likely real all along, she simply needed time to discover her true desires. Perhaps this provides final insight into her possible motive for setting the bombs.