This is index notation in order to express concisely a list of many equations or functions.
For example, the first line that you write states that there are two distinct functions γ1 and γ2 such that γ1 maps I1 into M where I1 is a subset of the real numbers and γ2 maps I2 into M where I2 is a subset of the real numbers.
The arrow notation defines the domain and codomain of the function: I1 is the domain of γ1 and M is the codomain.
The second set of notation has two indexes on each object; each index is taken to vary independently. For example, if the restrictions on α and β were explicitly given as α = 1, 2 and β = 1, 2, then the expression is a concise way of expressing the following list of expressions:
\text{For }\sigma_{11},\text{ let }J\sigma_{11} : V_{11}\rightarrow\text{Mat}_{n, n}(\Re)
\text{For }\sigma_{12},\text{ let }J\sigma_{12} : V_{12}\rightarrow\text{Mat}_{n, n}(\Re)
\text{For }\sigma_{21},\text{ let }J\sigma_{21} : V_{21}\rightarrow\text{Mat}_{n, n}(\Re)
\text{For }\sigma_{22},\text{ let }J\sigma_{22} : V_{22}\rightarrow\text{Mat}_{n, n}(\Re)
In words, the sentence defines a list of 4 functions Jσ_ab that maps each respective space V_ab into the set of nxn matrices with real components. In particular, this means that Jσ_12 takes an element of V_12 as an input and returns an nxn matrix with real components as an output. The explicit operation performed by the function on those elements of V_12 is not specified in this expression.
