As "layman's summary":
A proportional representation system, which correlates with an increase effective (in the sense of realistically electable) number of parties, correlates with increased turnout.
The raw number of parties correlates with increased turnout when the parties are sufficiently distinct from each other, e.g. as measured by party (platform) polarization.
The answer seems to be yes, but note that it's difficult to eliminate all potential confounders in cross-country research. A fairly cited 1990 paper:
This paper examines the record of western democracies to measure the impact of differing electoral formulae on the rate of voter turnout. The record of 509 national elections in 20 countries provides the basis for a regression analysis that clearly identifies higher turnout rates in PR systems that cannot be explained by a wide variety of control variables or traditional arguments about PR. The data also reveal a marked increase in electoral turnout over the last century.
Also, the precise measure used e.g. PR system [which is a binary (institutional) variable) vs. raw number of parties also seems to matter. (The effect is reportedly weaker of even contrary for the latter in some publications.) More recent research has looked at how polarization interacts with the number of parties in affecting turnout (for OECD countries):
Note that without the [parties x polarization] interaction term, number
of parties is insignificant in this [1.1] model, highlighting the poor explanatory power of this
variable without the interaction and being broadly in line with the literature. The controls
in Model 1.1 also behave as expected. The size of the population is significant and
negatively associated with turnout, while institutional indicators including proportional representation, compulsory voting, and unicameralism are all significant and positively
associated with turnout. [...]
Model 1.3 includes the interaction term between polarization and number of
parties. The interaction is significant and negatively associated with turnout, polarization
remains significant and positively associated with turnout, and number of parties becomes
significant and changes direction to be positively associated with turnout. [...]
polarization has a positive
effect upon turnout when there are few parties. This positive effect declines as the number
of parties increases. When there are more than four parties, polarization no longer has
an impact upon turnout. [...]
it is easy to imagine how the number
of parties would have little effect upon turnout when polarization levels are low. Systems
with low levels of polarization are unlikely to result in high voter turnout. The number
of parties in such a system simply makes little difference as, ultimately, there is little to
choose between the parties as they all inhabit a narrow ideological spectrum. The number
of parties only begins to have an effect upon turnout when polarization levels are high as
this can meaningfully affect the composition of government.
If you wonder on the technical details of how the latter paper measured polarization:
For the purpose of this article, an approach similar to Jansen, Evans, and Dirk de Graaf (2012) will
be utilized. The above authors’ measure of polarization was largely based on the standard deviation of party
ideology scores [taken from the Comparative Manifesto Project (CMP)]. This article differs from the above in that the standard deviation will be weighted by party size
in a similar fashion to Kim, Powell, and Fording (2010). [...] the use of manifesto data was chosen as it allows for comparison
over a prolonged period of time and is known to generate the smallest errors when predicting party ideology placements (Gable and Huber, 2000).