We investigate the unconstrained global optimization of functions with low effective dimensionality, that are constant along certain (unknown) linear subspaces. Extending the technique of random subspace embeddings in [Wang et al., Bayesian optimization in a billion dimensions via random embeddings. JAIR, 55(1): 361--387, 2016], we study a generic Random Embeddings for Global Optimization (REGO) framework that is compatible with any global minimization algorithm. Instead of the original, potentially large-scale optimization problem, within REGO, a Gaussian random, low-dimensional problem with bound constraints is formulated and solved in a reduced space. We provide novel probabilistic bounds for the success of REGO in solving the original, low effective-dimensionality problem, which show its independence of the (potentially large) ambient dimension and its precise dependence on the dimensions of the effective and randomly embedding subspaces. These results significantly improve existing theoretical analyses by providing the exact distribution of a reduced minimizer and its Euclidean norm and by the general assumptions required on the problem. We validate our theoretical findings by extensive numerical testing of REGO with three types of global optimization solvers, illustrating the improved scalability of REGO compared to the full-dimensional application of the respective solvers.