Dual-Tone Multi Frequency signalling
Essentially, the reason why you need two tones is to ensure that a normal human voice can't replicate the tones.  As such, by simultaneously generating a tone from a high-frequency group and low-frequency group, it is highly improbable that a human voice can replicate a sound from such differing ends of the frequency spectrum at the same time.  

If we were to adopt single tone dialing, wouldn't it be annoying if you were to talk to someone, your voice matches one of the dialing frequencies and it would randomly dial that number while talking?  It would certainly annoy me!

The Dual-Tone Multi Frequency tones (or DTMF for short) for a particular key is split up in the following grid:

1     |    2     |   3    |    A    |   697Hz
4     |    5     |   6    |    B    |   770Hz
7     |    8     |   9    |    C    |   852Hz
*     |    0     |   #    |    D    |   941Hz
--------------------------------------
1209Hz   1336Hz    1477Hz    1633Hz


For example, by pushing 1, you would generate two tones: 697 Hz and 1209 Hz from the low and high frequency ranges respectively.  In your post, you have already identified the frequencies used when dialing the number 3.  A,B,C,D are probably something you haven't seen before.  A,B,C,D were actually used as keys for the US Military's Autovon phone system.  They were used for:


A - Flash Override
B - Flash
C - Immediate
D - Priority


Pushing one of these keys gave your call priority with Flash Override being the highest.  The Autovon system has been disbanded for a long time, but many non-telephone applications still use those keys today.

One question that you may ask is how they decided to come up with these frequencies?  The frequencies were chosen to avoid harmonics - as in no frequency being a multiple of another.  Also, the difference between any two frequencies doesn't equal any of the other frequencies and the sum of any two frequencies isn't equal to any of the frequencies too.  The frequencies were generated within a +/- 1.5% tolerance of each other, the highest frequency must be as loud as the lowest frequency and as much as 4 dB louder.  This level difference is referred to as a "twist".