After winning the 1979 German national competition Jugend forscht in the field of mathematics and computer science, Speicher studied physics and mathematics at the Universities of Saarbrücken, Freiburg and Heidelberg. He received in 1989 his doctorate from Heidelberg University under the supervision of Wilhelm Freiherr von Waldenfels with thesis Quantenstochastische Prozesse auf der Cuntz-Algebra (Quantum Stochastic Processes on the Cuntz Algebra). From 2000 to 2010 Speicher was a professor at Queen's University in Kingston, Ontario. Since 2010 he is at the University of the Saarland.
In 2012, Speicher received the Jeffery–Williams Prize. He also received the Research Excellence Award of the President of Ontario. In 2014, he was an invited speaker with talk Free Probability and Random Matrices at the ICM in Seoul.
Speicher is married and has four children.