Why do we pray to Saints and Angels?
Well let me first start off by saying we do not pray to them like we do to God. We ask for their intercession. For example, the greatest Saint of all time in my opinion is Mary the Mother of God. She was sinless from conception much like Christ and is our great mediatrix. She is the greatest mediatrix because she is Christ's mother. She went through everything with Christ (the 7 swords of Mary) and she remained sinless while on her Christian pilgimage here on Earth. She is able to make our requests 'perfect' and without original sin that we are unable to. When you go to ask a favor of someone (and you are somewhat scared of what the answer would be) do you not ask for the intercession of a friend or go their mother to get them to phrase it in the 'perfect way'. That way when they answer, they are answering your favor/question but they do so with a smile on their face and without any hesitation because the person who interceded for you is perfect and the way they asked the question was perfect. After all Mary is full of grace by being The Holy Spirits spouse, God's daughter (the most perfect adorer of the blessed Trinity), and Mother of God (Jesus). It is all perfectly laid out in the Magnificat.
The other more orthox answer is the 'communion of saints'. Every Catholic should know these words at least, because we recite it in the Apostles Creed every weekend at church and everyday in our rosary. The communion of saints is quite simply the Church. "Since all the faithful form one body, the good of each is communicated to the others... We must therfore believe that there exists a communion of goods in the Church. But the most important member is Christ since he is the head...Therefore, the riches of Christ are communicated to all the mmbers, through the sacraments." ( The Catechism, 947). We also state we believe "in the resurection of the body" which because of Christ's death and ascention into Heaven is possible for us. That is one of the reasons why it is so important for us to realize Christ is the God-head.
We also pray to saints for their intercession because while they were on their Earthly pilgrimage they lived a life that would be considered pleasing to Christ (without sin as much as humanly possible) and performed at least one miracle while on Earth, which would lead us to believe that while they are in Heaven they can interced for us and bring our petition before the Lord and a possible miracle would occur. In the 54-day rosary novena to our Blessed Mother, we say during our intention before the rosary "I beg you to present my petition to your Divine Son. If you will pray for me, I cannot be refused. I know dearest Mother, that you want me to seek God's Holy Will concerning my request. If what I ask for should not be granted, pray that I may recieve that which will be of greater benefit to my soul". So, just because you are requesting something and it does not come to be, it is God's Will. As Jesus said during the agony in the garden when the Devil was tempting him and blood was running from every pour of his body at the vision of the torcher he was to undergo. He cried in anguish, "Abba! Father! if it be possible, let this chalice pass from me!" He also had to submit himself to the will of the Lord by saying these words "Not as I wilt, but as Thou wilt."
Now as far as praying to Angels, what I found is. Although Catholic prayers are mainly to God, church doctrine affirms that prayers can be made to angels asking for intercession. Catholic Answers Forums, the largest Catholic community on the internet writes, “The Church teaches us that we should pray not only directly to God, but also to those who are close to God, those who have the power to intercede upon our behalf. Indeed, we pray to the angels to help and watch over us..." The Catholic Encyclopedia also explains that it is biblical to pray to angels for intercession. In the apocryphal Book of Tobit 12:1, “The angel Raphael says: ‘I offered thy prayer to the Lord.’” It is common for a Catholic to ask an angel for help directly. Some Catholics ask a specific archangel to come to their aid.
I hope this helps.