That's a big promise, and many people who read about it think that it is a big leap to believe in.
I'd like to demonstrate that the leap is actually quite small.
What are the kinds of things that add a bit of clout to our prayers?
It helps if you pray out of right relationship with God. Praying this Psalm of repentance three times is a plea for mercy, and acknowledges the truth that we are sinners in need of God's forgiveness. We see our sin more clearly when we focus on the absolute absence of sin in Mary, the Mother of God. A heart in right relationship with God is more likely to be heard. 'The humble man's prayer pierces the clouds' Ecclesiasticus 35:17
Prayer with penance is more powerful than prayer alone. Praying for an hour is a sacrifice of all the other things we could be doing, and a challenge for anyone not used to spending an hour in prayer. Setting an hour aside to stay away from distractions is part of this too. Any hour of prayer is a reparative response to the question of Jesus on the night of His agony and betrayal, 'Could you not watch one hour with Me?' Matthew 26:40 Prayer flavoured with penance is more likely to be heard.
Praying in union with others is more effective than praying solo. Who could be a better prayer partner than Mary, the Mother of God? It is a mother's task to take the offerings of her children and arrange them in the way which will please the father most, removing the icky bits and adding her own special touch. A child might give mum a handful of flowers, and she will get rid of the insects, dead leaves and imperfect petals and add a matching ribbon and arrange them on a silver platter. With mum's help an ordinary offering becomes something especially pleasing. Think about how much more difficult it is to say No to two people who come before you with pleading eyes than to say No to one person. With Mary's help our prayers are more likely to be heard. Matthew 18:19
Praying in obedience is more meritorious than praying according to our own will. This Hour of Grace is prayed in accordance with heavenly instructions, and even if you feel a bit wobbly about the whether it is truly from heaven or not, you are still praying it in a way you wouldn't normally choose to. Romans 5:19
All four of these things are present when praying the Hour of Grace, so even without the promise attached, your prayers have a much greater hope of being heard and answered than usual.
For more information about the Hour of Grace, print and read this PDF:
Immaculate Mary, Mother of God, we entrust all the prayers of our hearts to you, that you may take them to your Son Jesus and obtain His smile of mercy upon them.