The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that $H$ is a representation of some Lie group $G$. So you want to understand this Lie group $G$, and generally you do it by looking at its Lie algebra. At least initially, this is understood as a way to make the problem of classifying Lie groups and their representations easier.


But there are Lie algebras which are not the Lie algebra of a Lie group, and people are still interested in them. One possible way to justify this perspective from a physical point of view is that Lie algebras might still be viewable as ("infinitesimal"?) symmetries of physical systems in some sense. However, I have never seen a precise statement of how this works (and maybe I just haven't read carefully enough). Can anyone enlighten me? 



What is the physical meaning of a Lie algebra symmetry?
I'm not sure which sort of examples of Lie algebras without the corresponding groups you have in mind, but here is a typical example from Physics.


Many physical systems can be described in a hamiltonian formalism. The geometric data is usually a symplectic manifold $(M,\omega)$ and a smooth function $H: M \to \mathbb{R}$ called the *hamiltonian*. If $f \in C^\infty(M)$ is any smooth function, let $X\_f$ denote the vector field such that $i\_{X\_f}\omega = df$. If $f,g \in C^\infty(M)$ we define their *Poisson bracket*
$$\lbrace f, g\rbrace = X\_f(g).$$
It defines a Lie algebra structure on $C^\infty(M)$. (In fact, a Poisson algebra structure once we take the commutative multiplication of functions into account.)


In this context one works with the Lie algebra $C^\infty(M)$ (or particular Lie subalgebras thereof) and not with the corresponding Lie groups, should they even exist. 


Symmetries in this context are functions which Poisson commute with the hamiltonian, hence the centraliser of $H$ in $C^\infty(M)$. They define a Lie subalgebra of $C^\infty(M)$.


**Added**


Another famous example occurs in two-dimensional conformal field theory. For example, the Lie algebra of conformal transformations of the Riemann sphere is infinite-dimensional: any holomorphic or antiholomorphic function defines an infinitesimal conformal transformation. On the other hand, the group of conformal transformations is finite-dimensional and isomorphic to $\mathrm{PSL}(2,\mathbb{C})$.


