Let's take the topic of prayer for an example of scripture proving scripture:
Jam 4:2 "Ye lust, and have not: ye kill, and desire to have, and cannot obtain: ye fight and war, yet ye have not, because ye ask not."
Jn 16:34 "Hitherto have ye asked nothing in my name: ask, and ye shall receive, that your joy may be full."
Mk 11:24 "Therefore I say unto to you, what things soever ye desire, when ye pray, believe that ye receive them, and ye shall have them."
Lu 18:1-8 "And he spake a parable unto them to this end, that men always ought to pray, and not to faint. Saying, there was in a city a judge, which feared not God, neither regarded man: and there was a widow in that city, and she came unto him, saying, Avenge me of mine adversary. And he would not for a while: but afterward thought within himself, though I fear not God, nor regard man; yet because this widow troubleth me, I will avenge her, lest by her continual coming she weary me. And the Lord said, hear what the unjust judge saith. And shall not God avenge his own elect, which cry day and night unto him, though he bear long with them? I tell you that he will avenge them speedily. Nevertheless when the Son of man cometh, shall he have faith on earth?"
Phil 4:6, 7 "Be careful for nothing; but in everything by prayer and supplication with thanksgiving let your requests be made unto God and the peace of God, which passeth all understanding, shall keep your hearts and minds in Christ Jesus."
Jam 5:16 "Confess your faults one to another, and pray one for another that ye may be healed. The effectual fervent prayer of a righteous man availeth much."
1 Jn 3:21, 22 "Beloved, if our heart condemn us not, then we have confidence toward God and whatsoever we ask, we receive of him, because we keep his commandments, and do those things that are pleasing in his sight."
Prayer that gets results begins with asking for a particular request. Is the prayer according to the Lord's will? Its using Jesus' name, our authority. Its striving to believe until the mustard seed of faith grows into a tree. Its maintaining a clear conscience by devotions in the Lord's word. Its wearing the Lord down with our continual coming to him with the prayer being requested. Its making those requests with thanksgiving in our heart. Its making the request with a fervent heart. Its persisting in striving to believe for that request as the Lord tries our faith that we might learn to persevere unto the obtaining of the promise.
So, scripture interprets scripture by taking the full council of God regarding a particular topic into account to obtain a clearer understanding of that topic; such as praying effectual prayers: in which all of the various scriptures, taken together are in agreement with ones viewpoint. And even then; 1 Cor 8:2 "And if any man think that he knoweth any thing, he knoweth nothing yet as he ought to know."
Maranatha!