Decibels and the Bode Asymptote
Replace the curve with two lines
The gold curve is the actual response, and the two green dashed lines are the asymptotes that stand in for it. As you change the frequency, the gap between the two at the present point is marked by a red bar; well above or below the cutoff they nearly touch, but in one place they spread farthest apart. Find that corner where the asymptotes bend, the cutoff frequency.
Decibels turn multiplying into adding
The decibel value is 20 times the common logarithm of the gain, dB = 20·log10|H|. Because it is a logarithm, multiplying gains becomes adding decibels, so when two stages are connected in series the total in decibels is just the sum of each stage's decibels. A gain of 1 (pass) is 0dB, a gain of 1/√2 (half power) is −3dB, a gain of 1/10 is −20dB, and a gain of 1/100 is −40dB.
On log axes the curve becomes straight
Put frequency on a logarithmic axis and both ends of a 1st-order response become straight lines. Well below the cutoff the gain is nearly 1, giving a horizontal line at 0dB, and well above it the gain falls by −20dB every time the frequency rises tenfold, a line of constant slope. These two lines are the asymptotes, and the point where they meet is the cutoff frequency. The actual curve rounds that corner smoothly.
At the corner it misses by exactly 3 dB
The asymptotes claim the response bends at 0dB at the corner, but there the actual curve has already dropped to −3dB. So the place where the straight-line approximation is worst is exactly the cutoff frequency, and the error there is exactly 3dB. One octave, a factor of two, away from the corner the error shrinks to 1dB, and farther out it is nearly 0. Draw just the two asymptotes and press the corner down by 3dB, and a hand-drawn Bode plot comes out almost identical to the real one.
Back to the first screen
While you changed the frequency, the gap between the actual curve and the asymptotes was nearly 0 far away, and it grew largest at the cutoff frequency, reaching 3dB. In other words, instead of drawing a curve it was enough to draw two straight lines and press the corner down by 3dB. Once decibels and a logarithmic axis have straightened out the curve, even a frequency response that looked complicated can be estimated quickly by hand.