The llncs class has its own way to define theorem-like environments and doesn't like to be called together with amsmath.
It already defines a proof environment, namely
\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
In a quite contrived way, this ensures that if \proofname is defined (possibly by a language definition file), it is used for labeling the proof.
Springer's house style seems to be of not generally adding a QED symbol; I checked Lang's “Algebra” and Hartshorne's “Algebraic Geometry”, which haven't QED symbols; to the contrary, Massey's “Singular Homology Theory” has Q.E.D. at the end of the line.
My suggestion is to load amsthm just for its proof environment and to not load thmtools, but to rely on Springer's \spnewtheorem command for defining your theorem-like environment (or using the predefined ones)
theorem case conjecture corollary definition example exercise
lemma note problem property proposition question solution remark
Load amsthm last and don't load thmtools.
\documentclass[runningheads]{llncs}
\usepackage{amsmath,amssymb}
\usepackage[algosection,ruled,lined,linesnumbered,longend]{algorithm2e}
\usepackage{hyperref}
\usepackage{cleveref}
\let\proof\relax\let\endproof\relax
\usepackage{amsthm}
\begin{document}
\title{Contribution Title\thanks{Supported by organization x.}}
\maketitle
\begin{abstract}
The abstract
\end{abstract}
\section{First Section}
\subsection{A Subsection Sample}
Please note
\begin{proof}
abc
\end{proof}
\end{document}
\let\proof\relax \let\endproof\relaxbefore\usepackage{amsmath,amsthm,amssymb}.