The reason we know God hears our prayers and uses them to accomplish His purposes is because He has promised it -- and God cannot lie. The Bible says, "This is the confidence we have in approaching God: that if we ask anything according to his will, he hears us. And ... we know that we have what we asked of him" (1 John 5:14--15).
Does this mean God always answers our prayers the way we wish He would? No, it doesn't; sometimes God's answer is "No" or "Wait" instead of "Yes." We see only part of the picture; God sees the whole, and He knows what is best for us (and for others). Sometimes this is hard for us to accept, but His will is always better than ours, and His ways are always better than ours.
Never forget, prayer is one of our greatest privileges, and it's only possible because Jesus Christ came to erase the barrier between us and God -- a barrier caused by sin. By His death and resurrection, Christ bridged the gap between us and God, and when we put our faith and trust in Him, we can bring all our burdens to God in prayer. The Bible says, "Let us then approach the throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need" (Hebrews 4:16).
Make sure of your relationship with Christ. Then make prayer part of your life every day, not only asking God to help others, but praising and thanking Him for all His blessings.