After saying سبحانك اللهم **in دعاء الاستفتاح - the opening supplication - we continue on and say وبحمدك, commonly translated as ‘and with your praise.’
There are a number of opinions on the implications of the letter و at the beginning of this phrase and how it affects the meaning of what we are saying, the most common of which would be ‘[I proclaim] Your Exaltedness while I am in the state of Your praise,’ - with the letter و signifying the state - or ‘[I proclaim] your Exaltedness and [I continue] with Your praise,’ with the letter و being a conjunction.
So after we have just affirmed and glorified Allah’s Exaltedness from every deficiency and negative attribute, we proceed to glorify and praise Him. And we praise Him with a very specific type of praise - the word حمد refers to a sincere, true form of praise that is given to one that is worthy and deserving of that praise, and that is accompanied by love.
So when we say وبحمدك what we are essentially saying is: After exalting You above all imperfections, I continue by praising you with my complete love, for You are worthy and deserving of all praise.