This should be helpful:

http://www.ssc.wisc.edu/sscc/pubs/MM/MM_TestEffects.html

Much software does not give the standard error of the variance of the intercepts and associated Wald p test as the distribution of the variance estimate is unlikely to be normally distributed ( most unlikely with a small estimate close to zero  and a small number of higher level units). The recommendation is to perform a Likelihood ratio test which has better properties - so compare the deviance of the models with and without the variance term. 

Many thanks Jochen Wilhelm. The link was useful. Following the article, I compared a model with only the fixed effects and a model with random effect as well with REML=FALSE. The P-value after LRT was

Well, "that the model with random effects is a better model" is your interpretation, and it depends on what you call "better". The low p-value of the LRT says that the ratio of the likelihoods of the data under the two models is quite unexpetedly large under the hypothesis that the random effects are zero. So the data are sufficient to (clearly) see that this hypothesis leads to a (restricted) model that is unable to describe the data as well as the unrestriced model. If this is what you call "better", then: yes.

But "better" can mean other things, like how accurate or how precise a model predicts the "boldness" of a new individual for a given "trial" etc. Such criteria can hardly be evaluated based on p-values.

In answer to your emailed question about LRT tests versus Wald tests with random parameters - there is a full discussion with references in Chapter 6  of this

https://www.researchgate.net/publication/260771330_Developing_multilevel_models_for_analysing_contextuality_heterogeneity_and_change_using_MLwiN_3_Volume_1_updated_March_2017_Volume_2_is_also_on_RGate 

Book Developing multilevel models for analysing contextuality, he...

Thank you very much @Kelvyn Jones.