Can a Levene Test be insignificant even if the variances of the two groups look different? My Levene test output is insignificant, but on visualising my data, the groups look like they have different variances.
How is this possible?
 A: The whole point of a statistical test is to determine whether two things that look different are more different than one would expect by chance, if they were actually identical.  So, the Levene's test is telling you: "these two groups look like they have different variances, but even if they had identical variances, they would look this different — or more different — about 25% of the time".
As @whuber suggests, maybe you are confused about what a Levene's test does? This picture doesn't directly quantify the variance, but it does show the interquartile range (the height of the purple boxes), which is closely related to the variance; the heights of the boxes are very similar even though the lefthand box is lower than the righthand box [indicating that there is a clear difference in the medians of the groups].
If you're thinking about using Levene's test to decide whether you should assume equality of variance when doing a t-test (i.e. using an ordinary vs a Welch t-test), you probably shouldn't do that:

A procedure once popular among researchers involves preliminary tests of equality of variances—for example, tests devised by Bartlett (1937), Levene (1960), Boos and Brownie (1989), and Brown and Forsythe (1974). If a preliminary test results in rejection
of the hypothesis of equal variances, one of the modiﬁed tests of location mentioned above or a related separate-variances test is performed. If the preliminary test does not result in rejection of $H_0$, homogeneity is assumed to be satisﬁed, and the usual two-sample Student t test of location is chosen. In recent years, many authors have discouraged the use of these preliminary tests.


When sample sizes are unequal, it appears that the most efficient strategy is to perform the Welch t test or a related separate-variances test unconditionally, without regard to the variability of sample values.

(Zimmerman 2004, emphasis added)
Zimmerman, Donald W. “A Note on Preliminary Tests of Equality of Variances.” British Journal of Mathematical and Statistical Psychology 57, no. 1 (2004): 173–81. https://doi.org/10.1348/000711004849222.
