Suppose we have an equation $\dot x=A(t)x$, further suppose $A(t)=\sum_{l=0}^{\infty} A_{l}\mathbf{1}_{[T_l,T_{l+1})}$, where $T_l$ for $l=0,1,2,\dots$ is a sequenece of time-instant and $A_l$ is a sequence of constant matrices. The fundamental matrix is given by then $$\mathcal{M}(t,t_0)=e^{(t-T_{l(t)}){A_{l(t)}}}\mathcal{M}(T_{l(t)},t_0)$$, where the index $l(t)$ is determined so that $t\in [T_{l(t)},T_{l(t)+1} )$.
I am having terrible confusion in understanding the index-notation $l(t)$, could anyone explain to me why it is used and what is its function here? Also, it would be great to learn how the fundamental matrix expression is becoming like that above. Thank you for your help. Ref: Page-5, here
Your equation with $\ l(t)\ $ is just another way of writing the equation $$ \mathcal{M}\big(t,T_0\big)=e^{\left(t-T_l\right)A_l}\mathcal{M}\big(T_l,T_0)\ \ \text{for}\ \ t\in\big[T_l,T_{l+1}\big)\ \ \text{and}\ \ l=0,1,\dots\ , $$ which is how you'd have to write it if you didn't use the function $\ l(t)\ $ (or some equivalent device) to simplify it.
For any $\ t\in\big[T_0,\infty\big)\ $, there's a unique value of $\ l\ $ such that $\ t\in\big[T_l,T_{l+1}\big)\ $, and $\ l(t)\ $ is just that unique value of $\ l\ $, which could be defined as $$ l(t)=\sup\big\{n\in\mathbb{N}\cup\{0\}\,\big|\,T_n\le t\,\,\big\}\ . $$ As you can see from a plot of the graph of $\ l(t)\ $ below, it's just a step function which picks out the proper value of $\ l\ $ to use in the equation above for any particular value of $\ t\ $.
As to the reason why the authors of the paper you cite write the equation with $\ l(t)\ $, as you have reproduced in your question, as opposed to how I've written it above, I can only guess, but if they quote the equation several times throughout the paper, then using the version with $\ l(t)\ $ avoids having to add the condition "$\ \text{for}\ \ t\in\big[T_l,T_{l+1}\big)\ \ \text{and}$$\ \ l=$$0,1,\dots\ $" every time, which would quickly become tedious if the number of times the equation is quoted is large.
The fundamental matrix $\ \mathcal{M}\big(t,T_0\big)\ $ reduces to the form given in these equations because $\ A(t)\ $ is piecewise constant. Over each of the intervals $\ \big[T_l,T_{l+1}\big)\ $, $\ A(t)=A_l\ $ is constant. Therefore, the solution of the matrix differential equation \begin{align} \dot{X}(t)&=A(t)X(t)\\ &=A_lX(t) \end{align} over the interval $\ \big[T_l,T_{l+1}\big)\ $ is $$ X(t)=e^{\left(t-T_l\right)A_l}X\big(T_l\big)\ . $$ The fundamental matrix $\ \mathcal{M}\big(t,T_0\big)\ $ is just the value of this solution when you start with $\ X\big(T_0\big)=I\ $, so you have $\ \mathcal{M}\big(T_0,T_0\big)=I\ $, $\ \mathcal{M}\big(T_l,T_0\big)=$$\,X\big(T_l\big)\ $ and $$ \mathcal{M}\big(t,T_0\big)=e^{\left(t-T_l\right)A_l}\mathcal{M}\big(T_l,T_0) $$ for $\ t\in\big[T_l,T_{l+1}\big)\ $.