Why should the electroweak Lagrangian have an $\rm SU(2)$ invariance? The QED Lagrangian has a $\rm U(1)$ invariance so as to preserve electric charge, which has been empirically demonstrated to be conserved.
The QCD Lagrangian has a $\rm SU(3)$ invariance so as to preserve the three colour charges, which have been empirically demonstrated to be conserved.
What are the two conserved quantities that justify the requirement that the Electroweak Interaction Lagrangian have a $\rm SU(2)$ invariance?
The three components of the vector in colour space for the quarks represent the wave function corresponding to the quarks being in different colour states.
What do the two-components in the Electroweak Lagrangian correspond to?
I understand that this $\rm SU(2)$ invariance should only regard the left chiral spinors.
 A: The SU(2) you are talking about is called weak isospin, and corresponds to conserved currents in the EW lagrangian, similarly to QED and QCD. As you said, its generators flip members of isodoublets to each other.
So, for instance, its $\tau^+$ acts on a left-handed electron and yields a left-handed electron neutrino. That is, the SU(2) doublets of the theory are $(\nu_e,e)^T$. Likewise, the left-chiral quarks fall into such doublets, $( u,d)^T$, etc...
The vacuum of the SM is in a funny SSB phase, and, unlike the lagrangian, is not invariant under that group, and so the charges corresponding to the currents are not quite well defined, and largely not conserved. (You could detect their shadow conservation ghostly poltergeists, if you are very careful, but let's not go there...)
To complicate matters, there is another group, a weak hypercharge U(1) which also couples fermions, and the SSB mixes it up with the 3rd isospin component of the above, in a beautiful mesh.
The $\tau^\pm$ pieces of the currents/charges were well-understood to describe β-decay at least a decade before the advent of the SM, by Feynman and Gell-Mann, something like terms  $W^+_\mu \bar\nu\gamma_\mu (1-\gamma_5)e$, etc, but the $\tau^3$ pieces arising in the commutators thereof with the hermitian conjugates appeared to specify interactions which were not there... until Glashow   unravelled their peculiar symmetry mixing structure... Weinberg & Salam organized them, and finally the corresponding "neutral current interactions were observed at Gargamelle, just as predicted.
A: The electroweak theory is “a bit” more complicate than this. I try to be super-synthetic and clear. The lagrangian is said to be invariant under SU(2) x U(1). The first is called Weak isospin (in analogy with the strong isospin, $e_L$ and $\nu_L$ are the analog of proton and neutron), the latter is the hypercharge. The left chirality Fermions are in a doublet representation while the right chirality fermions are singlet under SU(2). Up to this point all the particles must be massless since a term $m\bar{\psi}\psi$ will break the gauge invariance. Then to solve the problem of masses the higgs mechanism occurs providing the masses to fermions as well as to the gauge bosons. The mechanism is called Spontaneous symmetry breaking (often written as $SU(2)_I+ U(1)_Y\rightarrow U(1)_Q$. Namely the vacuum of the theory has not the full symmetry of the original lagrangian. It must be clear that SU(2) invariance is NOT only a matter of left chiral fermions, i. e. Gauge bosons ($W_\mu^I$) kinetic term is invariant under SU(2) and they transform with the adjoint representation of the group. A good introductory, not only, book to the Standard Model can be IMO Halzen and Martin’s “Quarks and Leptons”.
