Using staggered fermions and twisted boundary conditions, we calculate the K meson semileptonic decay vector form factor at zero momentum transfer. The HISQ formulation is used for the valence quarks, while the sea quarks are simulated with the asqtad action (MILC N_f=2+1 configurations). For the chiral and continuum extrapolation we use two-loop continuum CHPT, supplemented by partially quenched staggered CHPT at one loop. Our result is f_+textasciicircumKpi(0) = 0.9667+-0.0023+-0.0033, where the first error is statistical and the second is the sum in quadrature of the systematic uncertainties. This result is the first N_f=2+1 calculation with two lattice spacings and a controlled continuum extrapolation. It is also the most precise result to date for the vector form factor and, although the central value is larger than previous unquenched lattice calculations, it is compatible with them within errors. Combining our value for f_+textasciicircumKpi(0) with the latest experimental measurements of K semileptonic decays, we obtain textbarV_ustextbar = 0.2238+-0.0009+-0.0005, where the first error is from f_+textasciicircumKpi(0) and the second one is experimental. As a byproduct of our calculation, we obtain the combination of low-energy constants C_12textasciicircumr+C_34textasciicircumr-(L_5textasciicircumr)textasciicircum2 = (3.62+-1.00)x10textasciicircum-6.