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Assumption: You know your multiplication table reasonably well up to 10×10.
I am trying to explain this with an example,
Say you want to multiply - 16 x 13

• Step 1:  Add the unit’s digit of one to the other number –

                        Here, add 16 + 3 = 19
                       Or, add 13 + 6 = 19

• Step 2: Put a zero after the number (i.e. multiply it by 10)

                      Here, 19 becomes 190

• Step 3: Multiply unit’s digit of both the numbers

                     Here, 6x3 = 18

• Step 4: Add the product to the result of Step 2

                      Here, 190 + 18 = 208
                      Simple!! Isn’t it?

Another example, 17x19 =  (17+9)*10 + (7*9)  = 260 + 63 = 323

(10+a) * (10+b) = 100 + 10b + 10a + a*b
= (10+a+b) * 10 + a*b
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This can be extended to sums like 23 * 28
(20+a) * (20+b) = 400 + 20b + 20a + a*b
= (20+a+b) * 20 + a*b
So, 23*28 = (23+8)*20 + 3*8 = 620 + 24 = 644.