The Silence set out to kill the doctor to prevent him from ever answering his name on the Trenzalore, as something bad would obviously quite happen there, hence the phrase 'Silence will Fall'. Although it hasn't been revealed, based on deduction the Silence must have been trying to stop the Great Intelligence from getting into the Doctor's timeline and making all his greatest victories into failures, which would ultimately lead to the death of billions- if not trillions- of beings within the universe. Thus, it would seem plausible why the Silence didn't want the doctor to reach tranzalore to stop such a disaster from happening. HOWEVER, the Silence could only have set out on stopping the doctor from reaching Trenzalore after the events of the Name of the Doctor, meaning after the GI had already crossed into the Doctor's timestream- so basically they would want to create another paradox by preventing the GI from altering time by killing the doctor. Although if it was after the events from the Name of the Doctor, then surely they would have been aware that at the same time the GI goes and alters the Doctor's timeline, Clara jumps in moments after and most likely manages to prevent the Doctor's time stream from altering, meaning all those lives he did save remain saved. If this is so, then why did the Silence then need to go and and stop the Doctor from reaching Trenzalore, as the only plausible reason was to stop his name from being uttered in the Tardis- the one that becomes bigger after his death- which would open up the 'tracks of his tears'. But like I said, Clara stops the GI, so then why did the Silence need to go and stop this from happening as time still remains the same following the events of this episode.
It has not been said how aware the Silence were of the events that transpired on Trenzalore in "The Name of the Doctor". They may not even know of the Great Intelligence's and Clara's involvement in the matter, and could just be afraid of what the prophecy implies. In "The Wedding of River Song", the Doctor modeled the situation that he was in to Winston Churchill:
Doctor: "Suppose there was a man who knew a secret. A terrible, dangerous secret that must never be told. How would you erase that secret from the world? Destroy it forever, before it can be spoken."
Churchill: "If I had to, I'd destroy the man."
Doctor: "And silence would fall. All the times I've heard those words, I never realised it was my silence, my death. The Doctor will fall."
Considering this, it seems that the Silence's goal is not specifically to prevent the damage caused by the Great Intelligence corrupting the Doctor's timeline, but rather, to prevent his "terrible, dangerous secret" (which is now known to be the forgotten version of the Doctor portrayed by John Hurt) from being revealed.
WARNING MIGHT GET TOO CONFUSING
Good question in fact there are many good theories for this... here is my theory.
As we all know Clara Oswald is the "impossible girl" that means she well is supposed to do something impossible.The Doctor knows this that is why he chose her.So here is how to explain it.There are 2 dimensions, in one dimension a dude gets in a car accident. In another dimension the dude that was going to get in the car accidents sees himself get in a car accident, therefore he avoids it by not driving that day.Same with the Doctor. In one dimension the Doctor dies or silence will fall.So the Doctor by seeing his mistake he fixed it. Therefore in the other dimension the Doctor might have chosen the wrong companion and he failed when he/she was most needed this our dimension doctor might have foresaw it and chose Clara but since the silence did not have a T.A.R.D.I.S. they could not foresee it.