I won't explain the complete algorithm here. You can find a good explanation in Wikipedia or <PERSON> and <PERSON>
The aim of Grover search is to find some "marked" items ($|x_t\rangle$) in an unstructured database. The "marking" is done by a function called the oracle. The problem you are trying to solve is - 'Given an oracle can you find the items it marks in a database?'. The ability to implement the oracle doesn't imply that we can construct $|x_t\rangle$. And at times you will not even have the freedom to construct an initial state. For example when you use Grover search as an intermediate step in some other algorithm
It is not paradoxical. If you look at the intermediate steps closely all operations involved are unitary. So the entire algorithm is also unitary. The source of your confusion may be this: Initially $\langle \psi | x_t\rangle$ was small but the algorithm amplifies this value. 915
It is not a contradiction because unitary transformations preserve inner products if you apply them to both the vectors (i.e. rotation of the whole space). But here we change $| \psi \rangle$ and not $|x_t\rangle$. The amplification here happens the same way how the x-component of a 2D vector can change when you rotate it keeping the axes fixed.
Check out <PERSON> and <PERSON> for a really simple explanation. They effectively explain the whole thing in terms of two dimensional vectors.
An alternative way to think about it is using the <PERSON> equation. The Schrodinger Hamiltonian has two terms, the kinetic energy term and the potential energy term. If there is no potential then the tendency of the K.E term is to spread out the wave function. What <PERSON> search does is this: A potential is applied to the "marked" states and the system is allowed to evolve according to the Schrodinger Hamiltonian. The potential concentrates the amplitude at the marked states so that when you do a measurement there is a high chance that you will get these states. <PERSON> figured out his algorithm by thinking about it in this way and then discretizing <PERSON> equation. You can find his own explanation along these lines here 976