The Hell's Best Secret teaching is essentially a denouncement on what Ray Comfort calls "the modern Gospel", and a teaching on what he considers the proper use of Mosaic Law to prepare the heart for the Gospel.
Early on in the message, he speaks of a certain denomination's evangelism efforts during what they called "the decade of Harvest":
Let me make it more real for you. In 1991, in the first year of the
decade of harvest, a major denomination in the U.S. was able to obtain
294,000 decisions for Christ. That is, in one year, this major
denomination of 11,500 churches was able to obtain 294,000 decisions
for Christ. Unfortunately, they could only find 14,000 in fellowship,
which means they couldn’t account for 280,000 of their decisions, and
this is normal, modern evangelical results
According to Comfort, based on Church records that he's had access to, something like 80%-90% of those who have "been saved" later fall away. He blames this on the methods used to get people to come to Christ.
The "Modern Gospel" as he calls it, is one of life enhancement. Sinners are promised that God can heal their broken marriage, fix their drug problems, alcohol problems, and so forth. The "Modern Gospel" is, according to Comfort, a misguided attempt to make Christianity appealing by showing how much better the Christian life is than that of the unsaved.
He uses an illustration of a man on an airplane, who is given a parachute and is told that it would improve his flight, to illustrate how foolish this is.
Now listen to what the modern gospel says. It says, “Put on the Lord
Jesus Christ. He’ll give you love, joy, peace, fulfillment, and
lasting happiness.” In other words, “Jesus will improve your flight.”
So the sinner responds, and in an experimental fashion, puts on the
Savior to see if the claims are true. And what does he get? The
promised temptation, tribulation, and persecution. The other
passengers mock him. So what does he do? He takes off the Lord Jesus
Christ, he’s offended for the word’s sake (Mark 4:17), he’s
disillusioned and somewhat embittered, and quite rightly so. He was
promised peace, joy, love, fulfillment, and lasting happiness, and all
he got were trials and humiliation. His bitterness is directed toward
those who gave him the so-called “good news”. His latter end becomes
worse than the first: another inoculated and bitter backslider.
The rest of the teaching expands on the proper use of the Law, particularly the The pedagogical use (although he doesn't use that term in the teaching), which is to show sinners what sin is, to show them that they are, indeed, sinners, and show them their need of a Savior, and allow repentance. Without the Law, He argues (as does the Apostle Paul), sinners don't even know what sin is. How can they repent, if they don't know what sin is?
The teaching contains a section in which he illustrates using the Ten Commandments on an unrepentant sinner to show them their sinful nature in it's true light.
I say, “Ah, do you think you’ve kept the ten commandments?” He says,
“Ah, yeah…pretty much.” I say, “Let’s go through them. Ever told a
lie?” He says, “Ah, yeah…yeah, one or two.” I say, “What does that
make you?” He says, “A sinner.” I say, “No, no. Specifically, what
does it make you?” He says, “Well, man, I’m not a liar.” I say, “How
many lies, then, do you have to tell to be a liar? Ten and a bell
rings and ‘ppppbbbbtttt’ across your forehead? Isn’t it true if you
tell one lie, it makes you a liar?” He says, “Yeah…I guess you’re
right.” I say, “Have you ever stolen something?” He says, “No.” I say,
“Come on; you’ve just admitted to me you’re a liar.” I say, “Ever
stolen something, even if its small?” and he says, “Yeah.” I say,
“What does that make you?” He says, “A thief.” I say, “Jesus said, ‘If
you look at a woman and lust after her, you commit adultery with her
in your heart’ (Mat. 5:28).
Ever done that?” He says, “Yeah, plenty of times.” “Then from your own
admission, you’re a lying, thieving, adulterer at heart, and you have
to face God on judgment day; and we’ve only looked at three of the ten
commandments. There’s another seven with their cannons pointed at you.
Have you used God’s name in vain?” “Yeah…I’ve been trying to stop.”
“You know what you’re doing? Instead of using a four-letter filth word
beginning with ‘s’ to express disgust, you’re using God’s name in its
place. That’s called blasphemy; and the Bible says, ‘Every idle word a
man speaks he’ll give account thereofon the day of judgment’ (Mat.
12:36). ‘The Lord will not old him guiltless who takes his name in
vain’ (Ex. 20:7). The Bible says if you hate someone, you are a
murderer (1 John 3:15).”
The main point of the teaching is that instead of using promises of a better life to lure sinners to accepting Christ, we need to use God's law to show them their need for Christ, and drive them to the Savior.
As for the influence it's had in recent years:
While it's debatable whether Comfort's message is true, and it's not universally accepted, there's no denying that it has been widely accepted, and influential.
I personally first heard of the teaching at a Baptist Church, where a member gave me a cassette version of the audio teaching. I have since been presented with it from members of Assembly of God, Evangelical Free, and Independent Bible Churches. It's a teaching that's steadily gaining acceptance in a wide variety of denominations, although I've never seen it endorsed officially by any organization.
A few years back, Kirk Cameron (the actor) joined him in trying to spread the teaching, and as a result of Kirk's celebrity, the teaching has become even more popular.
The teaching has had influence in a number of well-known Christian movies, such as the Left Behind movies, Fireproof, and Courageous. In each of these movies, at least one character is seen using the same basic "have you ever told a lie" use of the law to speak to an unsaved person.
They've also been both on the radio and on Christian TV - with the Way of the Master radio, and The Way of the Master television show, and I've even heard the message preached on the ultra-conservative VCY America radio.