It is good idea to look at the beginning of Mishne Torah of the Rambam in order to get an idea of what the verse אין עוד מלבדו ["You were shown to know that the Lord is God, there are no others besides Him"] means. The way the Rambam explains it is this שכל הנמצאים צריכים לו והוא ברוך הוא אינו צריך להם ולא לאחד מהם...והוא שהתורה אומרת אין עוד מלבדו.
"For all things that exist need him, and He (blessed is He) does not need them, nor even one of them."
That is the idea is that things exist, but they need God in order to exist. There is no denial of the fact that things exist. But rather that their existence is dependent on God, while his existence is independent.
[This really ought not to be a surprise since the same idea can be found in any number of Musar books and books of the Geonim and Rishonim, e.g. אמונות ודעות (Faiths and Doctrines) of Saadia Gaon and the חובות לבבות (Obligations of the Heart)]
"For all things that exist need him, and He (blessed is He) does not need them, nor even one of them."
That is the idea is that things exist, but they need God in order to exist. There is no denial of the fact that things exist. But rather that their existence is dependent on God, while his existence is independent.
[This really ought not to be a surprise since the same idea can be found in any number of Musar books and books of the Geonim and Rishonim, e.g. אמונות ודעות (Faiths and Doctrines) of Saadia Gaon and the חובות לבבות (Obligations of the Heart)]