In the context of the lecture the point was that if one scrutinizes their own works for certainty of salvation, if one is truly honest with oneself, no work is good enough to meet the criteria of salvation. In fact, even our best efforts are tainted with sin. The only thing deep honest scrutiny of our works would produce is... despair. If you want to end up in despair like Luther before his evangelical breakthrough, take a good long look at your works.
I like how the Belgic confession puts it in Article XXIV:
Therefore we do good works, but not to merit by them (for what can we merit?); nay, we are indebted to God for the good works we do, and not He to us, since it is He who worketh in us both to will and to work, for his good pleasure. Let us therefore attend to what is written: When ye shall have done all the things that are commanded you, say, We are unprofitable servants; we have done that which it was our duty to do. In the meantime we do not deny that God rewards good works, but it is through His grace that He crowns His gifts.This sort of thing about works producing assurance is one of the reasons I'm not Puritan-crazy. I know, the Puritan writings are vast and varied. I'm not throwing them all out. I'm not particularly fond though of any that over-emphasize self-introspection to such an extreme that finding any sort of assurance of salvation is like finding a pearl of great price. Any sort of "assurance" that begins to function like a second blessing (the paradigm that sometimes works like, "Now I'm good enough to take communion") is not for me. I've run into these people. I met a dear old woman one time who explained to me she never took communion because she wasn't sure of her salvation.
Moreover, though we do good works, we do not found our salvation upon them; for we can do no work but what is polluted by our flesh, and also punishable; and although we could perform such works, still the remembrance of one sin is sufficient to make God reject them. Thus, then, we would always be in doubt, tossed to and fro without any certainty, and our poor consciences would be continually vexed if they relied not on the merits of the suffering and death of our Savior.
That Spurgeon quote certainly hits the target. I was curious about the particular quote attributed to him, so I did a cursory search, and came up with... nothing. I don't claim to be any sort of expert on Charles Spurgeon, nor would I even admit to having read many of his writings. I do though have many of his writings, either in electronic format, or actual books. I even took the time to contact the mastermind of the Spurgeon Archive, and he likewise could not find anything. So, my guess is the quote isn't Spurgeon's, or if it is, perhaps it's a summary of something he said.
As to assurance in general, here's my 2 cents. It's grounded upon the gracious promises of God in Christ. Yes, the witness of the Spirit in our hearts confirms it. Yes it is also confirmed by the way of obedience. But these later two do not trump the only foundation, the finished work of Christ and the promises of God. Someone focusing on the witness of the Spirit in the heart and the way of obedience demonstrated by works rather than the promises of God tends to be relying on personal experience for assurance.
Go ahead and scrutinize your works. Do you find yourself falling short of your duty? If you do, like I do, embrace the promises of God for your assurance. Do you want some sort of supernatural experience of the Holy Spirit assuring you of your faith? Cling to what the Holy Spirit wrote in the pages of Scripture. Believe what He says. You see, even the "experiential" assurances should lead you back to the promises of God for your assurance.