The ingenious circuit solution of Howland current source (aka "pump") was invented by Prof. Bradford Howland of MIT, about 1962 (see Fig. 5 on page 7 in the material below):

http://www.philbrickarchive.org/1964-1_v12_no1_the_lightning_empiricist.htm

It seems implausible, but almost 60 years later there are still no clear and simple explanations of this brilliant circuit. Formal ascertainments of WHAT was done predominate but they do not explain WHY it was done exactly in this way. In fact, extremely simple and intuitive explanations can be found… and they would be understandable even for a layman. Let's try it...

If we reduce the Howland electronic circuit to a simple equivalent electrical circuit (the first attached picture), we can see two cooperating current sources (basic and supplementary) connected in parallel to a common load. Thus we can easily discern the Howland's idea - to make a perfect current source by helping an imperfect current source.

The basic current source (on the left) is assembled by the voltage source V and the resistor R. It will produce the rated current I = V/R = IL only if the voltage across the load VL is zero (short connection). If there is some voltage (e.g. because the load has some resistance, capacitance, internal voltage), the effective “current-creating” voltage VR = V - VL across the resistor R decreases… аnd this decreases the current I = (V - VL)/R = V/R - VL/R.

The current has decreased by the value VL/R. It can be considered as erroneous current subtracted from the main current. So, if the same current is added (as if injected through a “pump”) to the main current, the rated current will flow through the load - IL = V/R - VL/R + VL/R = V/R… and it will not depend on the load voltage.

Obviously, the supplementary current IS = VL/R is produced by the second current source (on the right). It is assembled by a doubled voltage source 2VL and resistor with the same resistance Rs = R. Figuratively speaking, the right end of the resistor is always “lifted” with VL over its left end… so the “current-creating” voltage VRs = 2VL - VL = VL across the resistor RS stays equal to VL. As a result, the desired supplementary current IS = (2VL - VL)/R = VL/R, is added (“pumped” into the load).

In the real Howland circuit, the doubled voltage source 2VL is implemented as a non-inverting amplifier with a gain of 2 (the second picture). The circuit operation is visualized by voltage bars and current loops in the third picture. I created it in 2006 for my series of Circuit stories on the whiteboard:

https://www.circuit-fantasia.com/circuit_stories/understanding_circuits/current_source/howland_current_source/howland_current_source.htm