Start with the shortest instrument, whose yield is already a spot rate because it has one cash flow. Use it to discount the first coupon of the next bond, then solve for the rate that makes the rest of that bond's price work. Repeat outward.
Every discount curve, swap curve and valuation model sits on a bootstrapped set of spot-rates. Doing one by hand once is the fastest way to understand why coupon yields and zero yields are not the same thing.
Example: the 1-year spot is 4.80%. A 2-year 5% coupon bond trades at 100.86. Solve 100.86 = 5 / 1.048 + 105 / (1 + s2)^2. That gives s2 = 4.52%, the 2-year spot rate.
Related: spot-rate, zero-curve, par-yield-curve, forward-rate