Why A single-band resonant antenna acts as a crude bandpass filter: at the second or third harmonic of its design frequency the feedpoint impedance is far from a match, so harmonic energy from the transmitter is reflected rather than radiated. A multiband antenna is deliberately designed to accept and radiate on several bands, many of which (for example 7 MHz and 21 MHz) are harmonically related, so it happily radiates any harmonic output the transmitter produces. That makes good transmitter filtering and clean operation more important, since 97.307 still holds you responsible for spurious emissions.
Watch out Needing an antenna tuner is not the drawback being described, and many multiband designs such as trapped dipoles or fan dipoles present a usable match without one; open wire line is only required for certain designs like the doublet.
Multiband means multi-harmonic: the antenna that loves several bands will also radiate your harmonics.