The basicity of the double-bonded nitrogen is chiefly a result of the delocalization of charge permitted by the two single-bonded nitrogens with allylic lone pairs, as the resonance contributors you've drawn suggest. Notice that, upon protonation, the three nitrogens become chemically equivalent, with the positive charge shared equally among all three. If either of the single-bonded nitrogens is protonated, the resulting cation does not benefit from charge delocalization, which is a comparatively less stable arrangement.
Since the basicity is a function of the stability of the resulting guanidinium cation, it's a thermodynamic phenomenon. However, it may be worth mentioning that kinetic factors also likely favor protonation at the double-bonded nitrogen, since the lone pairs on the other two nitrogens are somewhat delocalized. This is the meaning of the $+M$ notation, which refers to the amine groups being electron-donating by the mesomeric effect (basically synonymous with resonance), while $-M$ would indicate groups which are electron-withdrawing. This is an effect active within the $\pi$-electron system of a molecule, which is probably better understood when considered from the standpoint of MO theory. The concept of resonance/mesomerism is a convenient abstraction that usually works well qualitatively to approximate the electron distribution in molecular orbitals (especially frontier orbitals). In reality, MO theory probably provides a more accurate rendering of the electron distribution, with $\pi$ orbitals that span multiple atoms. I say this to emphasize that resonance is not a physical process, but just a very expedient way of representing electron delocalization in $\pi$ orbitals.
Similary, $+I$ indicates a group being electron-donating by induction, while $-I$ denotes groups that are electron-withdrawing via induction. In essence, an electronegative group creates bond polarization, which can act to draw electron density away from an atom, while an electropositive group can do the opposite. This effect operates principally through the $\sigma$-bond skeleton of a molecule, and weakens as distance from the polarized group increases.