Church in Wales, I've been adopted by Cardiff
"For you did not receive a spirit of slavery to fall back into fear, but you have received a spirit of adoption. When we cry, ‘Abba! Father!’ it is that very Spirit bearing witness with our spirit that we are children of God,"
I interpret it as reinforcing Christ's ministry in fulfilling the Old Covenant and giving us the New - no longer was our relationship with God one of law, sacrifice and punishment, but that through his Incarnation, Christ came down to us in order to lead us back up to the Father as children of God, something we affirm when we cry 'Father'.
"Likewise the Spirit helps us in our weakness; for we do not know how to pray as we ought, but that very Spirit intercedes with sighs too deep for words."
We don't truly know how to pray - I notice that a lot of people pray in ways such as 'Please God, can I win the lottery' or 'Dear God, let England win the Euros this year', as though God is concerned with the trivial and wordly matters of our lives. Our prayers should solely be concerned with helping us to love our neighbour and to love God. But even then, if we pray for strength to live good lives, are we praying as we ought? God already knows the secrets of our hearts and to try and second guess him, as though he is not already acting in our lives, may not be right, either. The Holy Spirit acts in our lives in ways we cannot imagine, in ways we cannot predict ('too deep for words').
In short, we'll never really know how to pray, but we can try, and therefore instead of seeing prayer as a way to bring our situations to God and ask for help, prayer is an opportunity to connect with God, deepen our own awareness of the Spirit working in us, and separate our thoughts from those of worldly things. It reminds me of a prayer that centres me in that way:
Almighty God, the fountain of all wisdom,
you know our needs before we ask,
and our ignorance in asking:
have compassion on our weakness,
and give us those things
which for our unworthiness we dare not,
and for our blindness we cannot ask,
for the sake of your Son Jesus Christ our Lord.
This is just my take but I hope it helps in some way. Blessings.