What does the phrase 'background level' mean?
The phrase “background level” is generally used in an experimental context where you are measuring the effect on a variable of some agent or stimulus. You might expect that your control condition(s) — without the stimulus — would give you a measurement of zero, but often that is not the case, and one needs to subtract this control value — often referred to as ‘background’ — from the experimental value.

Such background values can be real — e.g. background radiation — or artefacts caused by fundamental limitations in the measuring system used. Generally the implication is that their origin is unknown or is not specific. That is why I think that the use of the term ‘background’ in the context of the lac operon is somewhat unfortunate. I would have used (and encourage you to use) the term ‘basal level’ instead*. As the text states, there is a low — but specific and necessary — expression of β-galactosidase in the non-induced situation, so that when lactose is added some of it can be converted to the inducer, allolactose, which will then result in the synthesis of the greater amount of β-galactosidase needed to deal with the rest of the lactose.

*Footnote 

In fact I hate the term ‘level’ except for water in the bath tub, and I would have used ‘amount’ or ‘activity’, but that’s not the point at issue here.