Saints are saints because of God, through the incarnation, crucifixion, resurrection and ascension of Jesus Christ. It is through his atoning work that saints live "though they die." We are able to have communion with them only because they have united themselves to Christ and rest in Him.
That said, because of this power through Christ, we pray to saints that they fulfill our petitions. The most basic petition, of course, would be that s/he intercede before Christ to grant us salvation and mercy. However, if these saints live in Christ, they are...well...alive. They are active beings and can work. They appear in visions, they heal the sick, raise the dead and work all sorts of miracles. I believe that it is indeed the saint, him or herself, that does that work. The saint works miracles. The distinction between the saint's and God's actions is that God works of his own power and authority, whereas the saint works only by the power and authority of God by virtue of their communion with Christ.
And, surely, this idea of God being "too busy" to answer prayer is seriously misguided and needs to be corrected.