Regarding DES
Suppose it takes a system 100 hours (actually, it may take you several weeks) unless you had the use of multiple (several hundred) computer systems all working on different ranges of keys. In other words, for DES, take one computer and test with keys from 0 to 1meg, computer 2, with keys from 1meg to 2 meg etc.
And if you could do that in 100 hours, great
E represents DES program = D
K represents the encryption/decription key
M represents your original input
C represents the Encrypted data
For DES E(k,m) = C
This says encrypt with DES, using the key k, against the message m, and generate the encrypted text C. To test that you can recover m using DES
D(k,c)=m = E(k,c) = m
DES says that the decryption program is the same as the encryption program
D(k,E(k,m))=m
(applying DES twice with the same key yields the original message)
3DES is the application of DES 3 times using the rule that k2 must not equal K1 or K3
3DES does the following.
E(k1,m) = c1
D(k2,c1) = c2
E(k3,c2) = c3
In words, use k1 to generate a result, then k2 on that result, and k3 and the second result. The D(k2,c1) applies the des algorithm in reverse, recall that D=E = DES.
Now lets suppose that with testing only with DES, it takes 100 hrs to discover k1 from c1 then it using c2 to discover k2, should take 10o hrs (which can only be done once you discover k1). The time now would be 100 x 100 or 10000 hrs, Simply because for every test with some value k, you need to do test with the result for step 2.
And now, knowing c3, it again takes 100hrs using c3 to discover k3 c2. Wwith triple DES, the amount of time to discover the keys is T**3, (T to the third power) if T=100 hrs, the time to discover 3DES would be 1000000 hrs. (41666 days, or 114 years)