Modular exponentiation is a well-known algorithm. It is routinely available in libraries and languages that can manipulate large integers, including Wolfram Alpha.
When making computations modulo a large number, one does not first make the whole computation in $\mathbb{N}$ and then take the remainder of the result, because for something like an exponentiation the intermediate result would be exceedingly large. Rather, the result is calculated with reductions modulo $n$ at each step. The basic idea for exponentiation is to write down the exponent $d$ in binary, and proceed one digit at a time, with the rules
$$ \begin{align*}
x^{2d} &= (x^d)^2 & \text{adding a binary digit \(0\) to the right of \(d\)} \\
x^{2d+1} &= (x^d)^2 \cdot x & \text{adding a binary digit \(1\) to the right of \(d\)} \\
\end{align*} $$
All computations are multiplications modulo $n$, it is never necessary to make computations involving larger numbers. There are variations on this algorithm, including iterating over the digits of $d$ from right to left instead of left to right. The number of multiplications is at most twice the number of digits of $d$.
There are techniques to perform a multiplication $a \cdot b \pmod n$ without calculating $a \cdot b$ which could be as large as $(n-1)^2$, the most popular being Montgomery reduction. Multiplication modulo $n$ can be done in time $O(\ell^2)$ where $\ell$ is the number of digits of $n$; it follows that exponentiation modulo $n$ can be done in time $O(\ell^3)$.
You can easily verify whether the value of $F(m)$ computed by Wolfram Alpha satisfies the equation $F(m)^d \equiv m \pmod n$ by running the modular exponentiation algorithm. (Easily with a computer, not by hand, for numbers of this size.)
We've seen how to compute $m^d \pmod n$ knowing $m$, $d$ and $n$. Solving the equation $F(m)^d = m \pmod n$ where $F(m)$ is the unknown is a different matter. Computing the $d$th root modulo $n$ is a hard problem in general, for which the only known methods are exponential in $\ell$ in the worst case. The problem is tractable (i.e. has polynomial complexity) when $n$ is prime, or more generally given a factorization of $n$.
It is apparent given the notations and the value of $e$ that these numbers are an example of an RSA key pair. An RSA key pair is generated starting with a number $n$ of the form $n = p \cdot q$ where $p$ and $q$ are (probable) primes and with a number $e$ satisfying certain conditions that I won't go into here; the knowledge of the factorization of $n$ makes it easy to compute a number $d$ such that $\forall m, (m^d)^e \equiv m \pmod n$ and $\forall m, (m^e)^d \equiv m \pmod n$. (If the factorization of $n$ was not known, then finding $d$ given $n$ and $e$ or equivalently finding $e$ given $n$ and $d$ would be a hard problem: it would be literally breaking the RSA cryptosystem.)
Given this supplementary knowledge, the $d$th root of $m$ modulo $n$ can be computed with a tractable operation: exponentiation. The solution $F(m)$ is $m^e \pmod n$.
The Python programming environment provides a convenient way of making calculations on large integers.
% python
Python 2.7.3 (default, Mar 13 2014, 11:03:55)
[GCC 4.7.2] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> m=112951412120151619251851615182016118201
>>> n=145906768007583323230186939349070635292401872375357164399581871019873438799005358938369571402670149802121818086292467422828157022922076746906543401224889672472407926969987100581290103199317858753663710862357656510507883714297115637342788911463535102712032765166518411726859837988672111837205085526346618740053
>>> d=89489425009274444368228545921773093919669586065884257445497854456487674839629818390934941973262879616797970608917283679875499331574161113854088813275488110588247193077582527278437906504015680623423550067240042466665654232383502922215493623289472138866445818789127946123407807725702626644091036502372545139713
>>> e=65537
>>> pow(m,e,n)
141781201173392662619041096248902274075236734827860244551641082388116267289894721928886528279400665340976615948053755946789302972467196231829204061441862114262117040263042768989461016645192295451421289297123899909172989906731037919155114663997141846344255396618911673987467404521083471325973763420396488985050L
>>> pow(pow(m,e,n),d,n)
112951412120151619251851615182016118201L
>>> pow(pow(m,e,n),d,n) == m
True
