A field $K$ is a $T_i$ field if a system of $m$ polynomial equations with no constant terms in $n$ variables has a non-trivial solution over $K$ if the degrees $d_k, k \in \{ 1, \dots, m \}$ of the polynomials $f_k$ satisfy the following condition $$n > d_1^i + \dots + d_m^i$$
I have a hard time understanding a part of the proof showing that a function field over a $T_i$ field is a $T_{i+1}$ field.
To do so, we consider a system of polynomial equations in $n$ variables with no constant terms over $K(X)$.
We then restrict ourselves to solutions that are polynomials in $K[X]$ with degree at most $s$. So far so good. We then claim that we can express the original indeterminates $X_i, i \in \{1,...,n\}$ over $K(X)$ as $$X_i = x_{i0} + x_{i1} X + x_{i2} X^2 + \dots + x_{is} X^s$$ where the $x_{ij}$'s are independent unknowns over $K(X)$. Apparently this is feasable because of an argument involving the transcendance degree of our fields. It seems to me as if we are just enlarging the number of variables but I don't see why this isn't always a valid procedure, i.e. why does one have to invoke transcendance degree ?
I would be thankful for any enlightening comment or help.
Edit: I added the proof I am referring to. See last line in parantheses. 
I am not sure what is the argument you are referring to as 'transcendence degree'.
But if I understand correctly, since as you have already observed, we are looking at solutions $(X_1,\dots, X_n)$ in $K[X]$ (rather than just in $K(X)$).
Now the $X_i$'s 'being' in $K[X]$, will have the form as a polynomial in $X$, where we treat the coefficients $x_{ij}$'s as new variables. To see how to show the existence of this variables:
This is my take on it: If the family $(x_{\nu\sigma})$ are algebraically dependent over $K(X)$, then $X_1,\dots, X_n$, (being algebraic combination of $x_{\nu\sigma}$ over $K(X)$), are algebraic dependent over $K(X)$, which is a contradiction, since by assumption $\{X_1,\dots, X_n\}$ is a transcendental basis for $K(X)(X_1,\dots, X_n)$ (so are algebraically independent over $K(X)$).
Ultimately we will be expanding the original system of polynomials in $X_i$'s into polynomials in $X$, the coefficients of those 'new' polynomials will give a new system of polynomials over $K$ in the variables $x_{ij}$'s. This is where we will apply that $K$ is $T_i$ field, and so on.