It means planting as much as you can in the smallest of plots. Learn to feed a family from the smallest of gardens. Plants are grown in compost rich soil Plus other amendments and fertilizers. You can use other soils also, which are explained below
But the part of growing tomatoes I like the best is that you can grow other plants with the tomatoes.
Some examples are: onion, carrots, beans, parsley and kale to name a few.

Intensive:
Planting types of plants that have compatably rooting systems. By that I mean if your main crop is tomatoes and you plant them deep you can also plant green onions which have a shallow root system and are harvested right at the time the tomatoes begin to flower. This is when the tomatoes need an extra boost of fertilizer to feed the new fruit and the onions won't be there to suck any of it up..

Square foot gardening:
Planting in measured plots. That is, plants are spaced as close as possible but leaving ONLY enough room for each root to get nutrients without stealing for the next plant. You have to learn each plants basic root ball diameter.

Containers come in many sizes and materials.
Barrels cut in half, old tires(also good for warm climate plants like melons) and pails make nice containers for tomatoes. Metal containers are ok but need a plastic liner or styrofoam inside to protect the roots from getting burnt or frozen on the metal. But the size and style is up to you. Just keep in mind that tomatoes require "at least" one square foot (1ft X 1ft and a minimum of 1ft deep) of soil.

The ideal container that I use is a wood frame about three and a half square foot.
That makes it easy to reach inside for weeding.
Then I line the inside sides with 2 inch thick styrofoam sheets cut to fit.
This protects the roots from quick temperature fluctuations.
I live in North Eastern Pennsylvannia and in the spring ( April or May ) temperatures can drop into the teens over night and reach the 80's in the daylight.
The styrofoam helps to keep the plants warmer and without a quick temperature change.
This will keep the plant from getting shocked and it will produce a tomato two weeks earlier.

One more important thing , the containers must be "sterile"
You can do this by washing or rinsing the containers with
"One cup of Clorox to One gallon of Warm Water"
Use plastic gloves to prevent from getting a reaction from the bleach.
And because of the dangerous fumes, it is recommended to do this outside.

Nitrogen:
When you overdue nitrogen fertilizer your plants will produce tall, beautiful green plants but with little or no fruit and flowers.
If they are not getting enough nitrogen you can spray them with a commecial foliar spray or some other source of nitrogen like diluted fish emultion and highly diluted Miracle Grow.
Also add some slow release nitrogen like Bonemeal or Fish meal to feed the soil.

Phosphorus:
If you use to much it will keep your plants from absorbing manganese, zinc and iron.
It also can take up to 2 years before it is washed out of the soil.
Add Rock Phosphate to your compost in the fall to correct a deficiency.
A more immediate result comes from speading a little bonemeal around the roots of the plant.
Be careful, not to much.

Potassium:
To much of this nutrient lasts for two years also.
A deficiency of potassium can be handled by adding wood ashes, manure or kelp materials to the soil. A weekly kelp foliar spray can help this problem more immediately.

More about fertilizer in myHomemade Sterile Soil Recipebelow.

Feed your soil and not your plants to have healthy Veggies.

Any fertile Soil will work but if you like to grow the older, less disease tolerant varieties, it is best to start with a Homemade Sterile Soil in your containers.
Sterile Soil sounds cold but it is the best way to keep the diseases at bay.
Below is an Old English Recipe that I like to use.