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I use the following code to manually draw arrows from one table cell to another.

documentclassbeamer
beamertemplatenavigationsymbolsempty
usepackageverbatim
usepackagetikz
usepackagepgfplots
usetikzlibraryarrows, arrows.meta, positioning
begindocument
beginframe[t]
frametitle
beginminipagetextwidth
begintable
begintabular*6c
15.1 & 23.7 & 19.7 & 15.4 & 18.3 & 23.0\
17.4 & 18.6 & 12.9 & 20.3 & 13.7 & 21.4\
10.3 & 26.1 & 15.7 & 14.0 & 17.8 & 33.8\
23.2 & 12.9 & 29.8 & 18.3 & 14.2 & 20.8\
13.5 & 17.1 & 20.7 & 27.1 & 18.9 & 16.6\
endtabular
endtable
beginminipagetextwidth
begintikzpicture[overlay]
draw<2-> [ultra thick, blue, latex'-] (4.1,1.66) -- +(0:.4);
draw<3-> [ultra thick, red, -latex'] (5.2,2.66) -- +(-74:1.4);
endtikzpicture
endminipage
endminipage
endframe
enddocument

Is it possible to automate drawing the arrows from one table cell to another, using cell references instead of manually drawing them?

Something like this?

documentclass[tikz]standalone
usetikzlibrarymatrix
begindocument
begintikzpicture
matrix (x) [matrix of nodes,row sep=1em,column sep=1em] %
15.1 & 23.7 & 19.7 & 15.4 & 18.3 & 23.0\
17.4 & 18.6 & 12.9 & 20.3 & 13.7 & 21.4\
10.3 & 26.1 & 15.7 & 14.0 & 17.8 & 33.8\
23.2 & 12.9 & 29.8 & 18.3 & 14.2 & 20.8\
13.5 & 17.1 & 20.7 & 27.1 & 18.9 & 16.6\;
draw[->] (x-2-1) -- (x-4-5);
% Node names: (<name of matrix>-<row>-<column>)
endtikzpicture
enddocument

In beamer, there may be some problems with matrix, as described here. Taking solutions from that question we have for example

documentclassbeamer
beamertemplatenavigationsymbolsempty
usepackageverbatim
usepackagetikz
usetikzlibraryarrows, arrows.meta, positioning, matrix
begindocument
beginframe[fragile]
frametitle
beginminipagetextwidth
centering
begintikzpicture
matrix (x) [matrix of nodes,row sep=1em,column sep=1em] %
15.1 & 23.7 & 19.7 & 15.4 & 18.3 & 23.0\
17.4 & 18.6 & 12.9 & 20.3 & 13.7 & 21.4\
10.3 & 26.1 & 15.7 & 14.0 & 17.8 & 33.8\
23.2 & 12.9 & 29.8 & 18.3 & 14.2 & 20.8\
13.5 & 17.1 & 20.7 & 27.1 & 18.9 & 16.6\;
draw<2-> [->] (x-2-1) -- (x-4-5);
endtikzpicture
endminipage
endframe
enddocument

Matrix is not really designed to draw a table, so drawing border lines is a bit tricky.

documentclass[tikz,margin=1mm]standalone
usetikzlibrarymatrix,calc
begindocument
begintikzpicture
matrix (x) [matrix of nodes,row sep=1em,column sep=1em] %
15.1 & 23.7 & 19.7 & 15.4 & 18.3 & 23.0\
17.4 & 18.6 & 12.9 & 20.3 & 13.7 & 21.4\
10.3 & 26.1 & 15.7 & 14.0 & 17.8 & 33.8\
23.2 & 12.9 & 29.8 & 18.3 & 14.2 & 20.8\
13.5 & 17.1 & 20.7 & 27.1 & 18.9 & 16.6\;
draw[->] (x-2-1) -- (x-4-5);
% Node names: (<name of matrix>-<row>-<column>)
foreach i in 1,...,6 
 draw ($(x-1-i.north west)+(-.5em,.5em)$) -- ($(x-5-i.south west)+(-.5em,-.5em)$);
foreach i in 1,...,5 
 draw ($(x-i-1.south west)+(-.5em,-.5em)$) -- ($(x-i-6.south east)+(.5em,-.5em)$);
draw ($(x-1-1.north west)+(-.5em,.5em)$) -| ($(x-5-6.south east)+(.5em,-.5em)$);
endtikzpicture
enddocument

The obligatory tikzmark and beamer-overlay-styles answer. It might be more useful if you use more features of table, which are harder to reproduce with a tikz matrix.

documentclassbeamer
beamertemplatenavigationsymbolsempty
usepackageverbatim
usepackagetikz
usetikzlibrarytikzmark,overlay-beamer-styles
usetikzlibraryarrows
begindocument
beginframe[t]
frametitle
begintable
begintabular*6c
15.1 & 23.7 & tikzmarknodem1319.7 & 15.4 & 18.3 & 23.0\
17.4 & 18.6 & 12.9 & 20.3 & 13.7 & 21.4\
10.3 & tikzmarknodem3226.1 & tikzmarknodem3315.7 & 14.0 & 17.8 & 33.8\
23.2 & 12.9 & 29.8 & tikzmarknodem4418.3 & 14.2 & 20.8\
13.5 & 17.1 & 20.7 & 27.1 & 18.9 & 16.6\
endtabular
endtable
begintikzpicture[remember picture,overlay]
draw[ultra thick, blue, -latex',visible on=<2->] (m13.south east) -- (m44.north west);
draw[ultra thick, red, -latex',visible on=<3->] (m33) -- (m32);
endtikzpicture
endframe
enddocument