Let us pretend that we have already constructed a continuous process $B: [0,1]\to\mathbb{R}$ with the same finite-dimensional distributions as a Brownian motion.
Keeping in mind that a stochastic process is a family of uncountably many random variables $\omega\mapsto B(t,\omega)$ for fixed $t$,
take a sequence $B_0\text{, }B_1\text{, }\ldots$ of independent $\mathscr{C}[0,1]$-valued random variables (i.e. continuous random variables on $[0,1]$) with the distribution of the Wiener process, and define $\{B(t):t\geq0\}$ by gluing together the parts, more precisely by $$B(t)=B_{\lfloor t \rfloor}(t-\lfloor t \rfloor)+\sum\limits_{i=0}^{\lfloor t \rfloor-1}B_i(1)\text{, for all }t\geq0\tag{1}$$ This defines a continuous random function $B:[0,\infty)\to\mathbb{R}$ and one can see easily that this is a standard Brownian motion.
I really cannot understand the logic underpinning the passage from $\mathscr{C}[0,1]$ to $\mathscr{C}[0,+\infty]$, that is how to get from a Brownian motion on $\mathscr{C}[0,1]$ (that is, the one we pretended to have already constructed) to a Brownian motion on $\mathscr{C}[0,+\infty]$ (that is, $(1)$).
How to get to $(1)$ and why has $(1)$ exactly that form?