= Solution
Let $X$ be an <affine variety> over an <algebraically closed field> $k$, with reduced <coordinate ring> $A=k[X]$. On an open subset $U$, define $\mathcal O_X(U)$ to be the functions $h:U\to k$ which, near every point, have an expression $a/b$ with $a,b\in A$ and $b$ nowhere zero on that neighbourhood. Restriction is ordinary restriction of functions. Compatible local functions glue uniquely and remain locally quotients, so this defines the <structure sheaf>. On the <principal open subset> $D(f)$ it gives $A_f$, and at $P$ it gives the <local ring> $A_{\mathfrak m_P}$. The principal-open identification can also be seen by identifying $D(f)$ with the affine algebraic set obtained by adjoining a coordinate $t$ and the equation $tf=1$; its <coordinate ring> is $A[t]/(tf-1)\cong A_f$. This construction permits several <irreducible components>; no division by a function vanishing identically on a component is assumed.
An abstract <algebraic variety> is a <locally ringed space> locally isomorphic to such affine models, with a finite affine cover; we use the usual separated convention. Reducible <varieties> are allowed here. The separation condition says that the <diagonal morphism> is a <closed immersion>. Two <varieties> are related by a <birational map> with an inverse when some dense open subset of one is <isomorphic> as a locally ringed space to a dense open subset of the other.
Define the <ring of rational functions on a reduced variety> by pairs $(U,h)$, where $U\subseteq X$ is dense open and $h\in\mathcal O_X(U)$, identifying two pairs when their restrictions agree on a common dense open subset. Finite intersections of dense open subsets are dense, so addition and multiplication on these intersections are well defined. For an <irreducible variety> this gives the usual <function field>; for a reducible <variety> it need not be a <field>. Restriction to any dense open subset $V$ induces an <isomorphism>
$$
\operatorname{Rat}(X)\cong\operatorname{Rat}(V).
$$
Indeed, intersecting domains with $V$ defines the restriction, and a dense open subset of $V$ is also dense open in $X$, which defines its inverse. An <isomorphism> between dense opens therefore induces an <isomorphism> of their rational rings. Thus \b[birational <varieties> have isomorphic rings of rational functions], also in the reducible case.
For the finite-cover assertion, write the closed complement of $U$ as $V(I)$, where $I=(f_1,\ldots,f_r)$ is an <ideal> of the <Noetherian ring> $A$. Then
$$
U=D(f_1)\cup\cdots\cup D(f_r).
$$
Let $\mathfrak p_1,\ldots,\mathfrak p_t$ be the <minimal prime ideals> of $A$, corresponding to its <irreducible components>. Since $U$ is dense, it meets every component, so $I$ is contained in none of these primes. <Prime avoidance> supplies $s\in I$ outside their union. Consequently $D(s)\subseteq U$ meets a dense open subset of every component and is dense in $X$. Add $D(s)$ to the finite cover if necessary. This proves that \b[a finite principal-open cover can be chosen with a dense member]. It does not say that every principal-open cover already has such a member: for $X=V(xy)$, the dense open $X\setminus\{(0,0)\}=D(x)\cup D(y)$ has neither displayed member dense, although adding $D(x+y)$ supplies a dense member.
For clarity, the <zero divisors> in this reduced <Noetherian ring> are exactly $\bigcup_i\mathfrak p_i$. If $ab=0$ and $a$ avoids every $\mathfrak p_i$, reduction in the domain $A/\mathfrak p_i$ forces $b\in\mathfrak p_i$ for all $i$, hence $b=0$. Conversely, if $a\in\mathfrak p_i$, choose for every $j\ne i$ an element $b_j\in\mathfrak p_j\setminus\mathfrak p_i$, possible because distinct minimal primes are incomparable. Their product $b$ is not in $\mathfrak p_i$ but belongs to all the other primes. Then $b\ne0$ and $ab\in\bigcap_j\mathfrak p_j=0$. For a single minimal prime, reducedness makes that prime zero and this argument uses $b=1$. It follows that
$$
s\text{ is a non-zero-divisor}\quad\Longleftrightarrow\quad D(s)\text{ is dense}.
$$
The <total quotient ring> $S^{-1}A$, where $S$ consists of these <non-zero-divisors>, maps to $\operatorname{Rat}(X)$ by
$$
a/s\longmapsto\text{the class of the regular function }a/s\text{ on }D(s).
$$
The equivalence relation for <localization> makes this a well-defined $k$-algebra homomorphism. For surjectivity, a rational class represented on $U$ restricts to the dense principal open $D(s)\subseteq U$ constructed above. Its section there belongs to $A_s$, so it is $a/s^r$ and lies in the image. For injectivity, if such a fraction is zero on a dense open subset, its numerator vanishes on that dense subset. The numerator's zero set is closed, so it vanishes everywhere on $X$ and is zero in the reduced <coordinate ring>. Thus
$$
\boxed{\operatorname{Rat}(X)\cong S^{-1}A.}
$$
In particular $A\to S^{-1}A$ is canonically <injective>: $a/1=0$ implies $sa=0$ for some <non-zero-divisor> $s$, hence $a=0$.
Finally let $Y$ be the reduced closed union of exactly those <irreducible components> of $X$ which contain $P$. Removing all the other components gives an open neighbourhood of $P$ on which $X$ and $Y$ coincide, including their <structure sheaves>. Hence $\mathcal O_{X,P}\cong\mathcal O_{Y,P}$. Write $B=k[Y]$ and let $\mathfrak n_P$ be its maximal ideal at $P$. Every minimal prime of $B$ is contained in $\mathfrak n_P$, because every component of $Y$ contains $P$. If $b\notin\mathfrak n_P$, it avoids each such prime and is a <non-zero-divisor>. Therefore the map of <localizations>
$$
B_{\mathfrak n_P}\longrightarrow Q(B)=\operatorname{Rat}(Y),
\qquad a/b\longmapsto a/b,
$$
is defined and <injective>: if its value is zero, a <non-zero-divisor> annihilates $a$, so $a=0$. We obtain the desired <local ring embeds in rational functions on incident components>:
$$
\boxed{\mathcal O_{X,P}\hookrightarrow\operatorname{Rat}(Y),
\qquad Y=\bigcup_{P\in X_i}X_i\ \text{with reduced structure}.}
$$
At an intersection point one must retain all incident components. For example, the <local ring> of $V(xy)$ at the origin has two nonzero germs $x,y$ with $xy=0$; it cannot embed into the <function field> of just one of its axes.
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